Smallest Cubic Crossing Number Graph

Author

Eric W. Weisstein
February 22, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/SmallestCubicCrossingNumberGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SmallestCubicCrossingNumberGraph.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

References

https://www.mathematica-journal.com/data/uploads/2009/11/CrossingNumberGraphs.pdf

https://oeis.org/A110507

Edge insertion/excision

EdgeInsert[g_System`Graph,{u1:UndirectedEdge[a_,b_],u2:UndirectedEdge[c_,d_]}]:=Module[{n=VertexCount[g],v=GraphEmbedding[g]},SetProperty[EdgeAdd[VertexAdd[EdgeDelete[g,{u1,u2}],{n+1,n+2}],UndirectedEdge@@@{{a,n+1},{b,n+1},{c,n+2},{d,n+2},{n+1,n+2}}],VertexCoordinates->Join[v,Mean/@{v[[{a,b}]],v[[{c,d}]]}]]]

EdgeInsert[g_System`Graph,l_List]:=Fold[EdgeInsert,g,l]

EdgeExciseWrong[g_System`Graph,u:UndirectedEdge[a_,b_]]:=Module[{v=VertexList[g]},
System`Graph[DeleteCases[v,b],Union[Sort/@(DeleteCases[EdgeList[g],u]/.b->a)],VertexCoordinates->Delete[GraphEmbedding[g],b],VertexLabels->"Name"]
]

EdgeExcise[g_Graph,e_UndirectedEdge]:=Module[{h=EdgeDelete[g,e],p},
p=Flatten[Position[VertexDegree[h],2]];
VertexDelete[EdgeAdd[h,UndirectedEdge@@@(Flatten[Position[#,1]]&/@Normal[AdjacencyMatrix[h][[p]]])],p]
]/;CubicGraphQ[g]

Smallest Cubic Graphs with CN == n

data=Monitor[GraphData[g=#,{"CN","V","#"}]&/@GraphData[{"Connected","Cubic"}],g];//Timing

tabledata=SplitBy[Sort[data],{First,#[[2]]&}]

Length/@Most[tabledata[[All,1]]]

Having trouble getting 37 for ({{37, 90, ({{{"Cubic",{90,1}}}})}})

Having trouble getting 56 for ({{56, 70, ({{"Balaban10Cage"}})}})

Having trouble getting 56 for ({{56, 70, ({{"HarriesGraph"}})}})

TextGrid[smallest=Append[Take[#[[1]],2],Column@#[[All,-1]]]&/@Most[tabledata[[All,1]]],Dividers->All,Alignment->{{{Decimal},Left},Top}]

Row[TextGrid[#,Dividers->All,Alignment->{{{Decimal},Left},Top}]&/@Partition[smallest,UpTo[22]],"   ",Alignment->{Automatic,Top}]

Cubic CN graphs

0

CrossingNumber[GraphData["TetrahedralGraph"]]

1

CrossingNumber[GraphData["UtilityGraph"]]

2

TextGrid[Sort[data=GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===2&]],Dividers->All]

(rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->50]])//Timing

3

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===3&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->50]]

4

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===4&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

5

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===5&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

6

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===6&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

7

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===7&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

8

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===8&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

9

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===9&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

10

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===10&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

11

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===11&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

12

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===12&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->200]]

13

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===13&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

14

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===14&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

15

TextGrid[data=Sort[GraphData[#,{"V","#","CN","RCN"}]&/@Select[GraphData[{"Connected","Cubic"}],GraphData[#,"RCN"]===15&]],Dividers->All]

rules=Thread[data[[All,2]]->CrossingNumber[GraphData/@data[[All,2]],MaxIterations->100]]

170 (conjectured)

CrossingNumber[GraphData["Tutte12Cage"],MaxIterations->1]//Timing

CrossingNumber[GraphData["Tutte12Cage"],MaxIterations->10]//Timing

CrossingNumber[GraphData["Tutte12Cage"],MaxIterations->100]//Timing

CrossingNumber[GraphData["Tutte12Cage"],MaxIterations->10^3]//Timing

Figure

cgdata=SortBy[DeleteCases[{#,Flatten[StringCases[Cases[GraphData[#,"StandardNames"],_String],"CrossingNumberGraph"~~c__:>c]],GraphData[#,"CN"]}&/@Complement[GraphData["Cubic"],GraphData["Refuted*"]],{_,{},_}],Reverse]

GraphicsGrid[DeleteCases[
Map[(If[(g=GraphData[#[[1]],"Graph","MinimalCrossing"])==={}, g={GraphData[#[[1]]]}];Show[g[[1]]//StyleGraphs,PlotLabel->TextCell[Column[{"CNG "<>#[[2]],If[StringQ[#[[1]]]&&StringMatchQ[#[[1]],"Crossing*"],"",StringReplace[GraphData[#[[1]],"Name"],{"number graph "->""}]]},Alignment->Center],PageWidth->125,TextAlignment->Center],BaseStyle->Directive[Black,FontFamily->"Times",Italic]])&,Partition[cgdata,UpTo[5]],{2}],{}],ImageSize->700,Alignment->Top,Spacings->{0,50}]

Construction

K7

ShowGraph[g=//ToCombinatoricaGraph]

RecognizeGraph[g]

CrossingNumber[g]

Append[Most[GraphData["K7","Embeddings"]],ToCommonEdges[g,Graph["K7"]]//Round]

GraphData["K7","AllImages"]

4A == MoebiusKantorGraph

ShowGraph[g=//ToCombinatoricaGraph]

r=RecognizeGraph[g]

GraphData[r,"StandardNames"]

GraphData[r,"AllImages"]

Append[Most[GraphData[r,"Embeddings"]],ToCommonEdges[g,Graph[r]]//Round]

CrossingNumber[g]

5A == PappusGraph

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

GraphData[r,"StandardNames"]

CrossingNumber[g]

GraphData[r,"AllImages"]

ToCommonEdges[g,Graph[r]]

Append[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]//Round]

5B

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

CrossingNumber[g]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]//Round]

6B

r=RecognizeGraph[g=ToCombinatoricaGraph[g=]]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]//Round]

CrossingNumber[g]

6C

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

CrossingNumber[g]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]//Round]

7A

r=RecognizeGraph[g1=ToCombinatoricaGraph[]]

r=RecognizeGraph[g2= ToCombinatoricaGraph[]]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g2,Graph[r]]//Round]

CrossingNumber[g]

7B [OLD7B; wrong; deleted]

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

CrossingNumber[g]

RawGraphData["CrossingNumberGraph7B","Name"]:="7-crossing number graph B"
RawGraphData["CrossingNumberGraph7B","AlternateStandardNames"]:={}
RawGraphData["CrossingNumberGraph7B","AlternateNames"]:={}
RawGraphData["CrossingNumberGraph7B","CanonicalForm"]:={{1879048192},{-2147446784},{-2147480576},{-2147459072},{5505024},{11010048},{787456},{3147776},{67178496},{134356992},{83894272},{151027712},{100696064},{167788544},{4199424},{8407040},{1075314688},{268763136},{274726912},{1082261504},{553713664},{570556416}}
RawGraphData["CrossingNumberGraph7B","Information"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","Notation"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","ConnectedComponentIndices"]:={{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}}
RawGraphData["CrossingNumberGraph7B","ConnectedComponentCount"]:=1
RawGraphData["CrossingNumberGraph7B","ConnectedComponentGraphNames"]:={"CrossingNumberGraph7B"}
RawGraphData["CrossingNumberGraph7B","VertexCount"]:=22
RawGraphData["CrossingNumberGraph7B","Embeddings"]:={{{0.959,0.282},{-0.142,0.99},{-0.655,0.756},{-0.655,-0.756},{-0.142,-0.99},{0.415,0.91},{0.959,-0.282},{0.655,-0.756},{0.841,0.541},{-1.,0.},{-0.841,0.541},{-0.841,-0.541},{0.142,-0.99},{0.142,0.99},{-0.415,0.91},{-0.415,-0.91},{1.,0.},{0.655,0.756},{-0.959,0.282},{-0.959,-0.282},{0.841,-0.541},{0.415,-0.91}},{{0.959,0.282},{0.142,0.99},{0.655,-0.756},{-0.841,0.541},{-0.415,0.91},{-1.,0.},{0.142,-0.99},{-0.841,-0.541},{0.841,0.541},{0.415,0.91},{0.841,-0.541},{-0.415,-0.91},{1.,0.},{-0.142,0.99},{0.415,-0.91},{-0.655,0.756},{-0.142,-0.99},{-0.959,0.282},{0.655,0.756},{-0.655,-0.756},{-0.959,-0.282},{0.959,-0.282}},{{0.959,0.282},{-0.841,-0.541},{-0.415,-0.91},{0.142,-0.99},{0.959,-0.282},{0.655,-0.756},{-0.655,0.756},{-0.142,0.99},{0.841,0.541},{-0.959,-0.282},{0.415,0.91},{-0.959,0.282},{1.,0.},{0.841,-0.541},{-0.655,-0.756},{-0.142,-0.99},{-0.841,0.541},{0.415,-0.91},{0.655,0.756},{-1.,0.},{-0.415,0.91},{0.142,0.99}},{{0.959,0.282},{-0.415,-0.91},{-0.841,-0.541},{0.655,-0.756},{0.959,-0.282},{0.142,-0.99},{-0.655,0.756},{-0.959,0.282},{0.841,0.541},{0.415,0.91},{-0.959,-0.282},{-0.142,0.99},{1.,0.},{-0.142,-0.99},{-0.655,-0.756},{0.841,-0.541},{-0.415,0.91},{0.415,-0.91},{0.655,0.756},{0.142,0.99},{-0.841,0.541},{-1.,0.}},{{0.959,0.282},{-0.959,0.282},{-0.142,-0.99},{0.142,0.99},{0.655,0.756},{-0.959,-0.282},{-0.655,-0.756},{0.655,-0.756},{-0.415,0.91},{-0.841,0.541},{0.142,-0.99},{0.959,-0.282},{0.841,0.541},{-1.,0.},{-0.415,-0.91},{0.415,0.91},{1.,0.},{-0.142,0.99},{-0.655,0.756},{0.841,-0.541},{-0.841,-0.541},{0.415,-0.91}},{{0.959,0.282},{-0.841,0.541},{-1.,0.},{0.142,0.99},{0.655,0.756},{-0.415,0.91},{-0.142,-0.99},{-0.655,-0.756},{1.,0.},{0.841,-0.541},{-0.959,-0.282},{0.415,-0.91},{0.841,0.541},{-0.655,0.756},{-0.959,0.282},{0.415,0.91},{0.142,-0.99},{-0.142,0.99},{0.959,-0.282},{0.655,-0.756},{-0.415,-0.91},{-0.841,-0.541}},{{0.959,0.282},{-1.,0.},{-0.841,0.541},{-0.415,-0.91},{-0.841,-0.541},{0.841,-0.541},{0.415,-0.91},{0.415,0.91},{1.,0.},{-0.142,0.99},{-0.655,0.756},{-0.142,-0.99},{0.841,0.541},{-0.959,-0.282},{-0.959,0.282},{-0.655,-0.756},{0.142,-0.99},{0.959,-0.282},{-0.415,0.91},{0.142,0.99},{0.655,-0.756},{0.655,0.756}},{{0.959,0.282},{0.415,-0.91},{0.142,0.99},{-0.655,-0.756},{-0.959,-0.282},{-0.142,-0.99},{-0.415,0.91},{-0.841,0.541},{0.841,0.541},{0.655,-0.756},{0.415,0.91},{0.959,-0.282},{-1.,0.},{0.142,-0.99},{-0.142,0.99},{-0.841,-0.541},{1.,0.},{-0.415,-0.91},{0.655,0.756},{0.841,-0.541},{-0.655,0.756},{-0.959,0.282}},{{0.959,0.282},{0.655,-0.756},{0.142,0.99},{-0.655,-0.756},{-0.415,0.91},{-0.841,0.541},{0.959,-0.282},{-1.,0.},{-0.142,-0.99},{0.415,-0.91},{0.415,0.91},{-0.841,-0.541},{0.841,0.541},{-0.655,0.756},{0.841,-0.541},{-0.142,0.99},{1.,0.},{-0.415,-0.91},{0.142,-0.99},{-0.959,-0.282},{-0.959,0.282},{0.655,0.756}},{{0.959,0.282},{-0.841,0.541},{-0.142,-0.99},{-0.655,-0.756},{0.655,0.756},{0.142,0.99},{-0.415,0.91},{0.655,-0.756},{-0.959,-0.282},{-0.959,0.282},{0.142,-0.99},{0.959,-0.282},{0.841,0.541},{0.415,0.91},{-0.655,0.756},{-0.415,-0.91},{1.,0.},{-0.841,-0.541},{-1.,0.},{0.841,-0.541},{-0.142,0.99},{0.415,-0.91}},{{0.959,0.282},{-0.959,0.282},{-0.142,0.99},{0.415,0.91},{-0.142,-0.99},{-0.655,-0.756},{-0.959,-0.282},{0.655,-0.756},{0.841,0.541},{-0.841,0.541},{-0.415,0.91},{0.959,-0.282},{0.142,-0.99},{-0.415,-0.91},{-1.,0.},{0.142,0.99},{1.,0.},{0.655,0.756},{-0.655,0.756},{0.841,-0.541},{-0.841,-0.541},{0.415,-0.91}},{{0.959,0.282},{0.142,0.99},{-0.415,-0.91},{-0.841,0.541},{-1.,0.},{-0.415,0.91},{0.142,-0.99},{0.655,-0.756},{0.841,0.541},{0.415,0.91},{-0.655,-0.756},{0.959,-0.282},{-0.959,-0.282},{-0.142,0.99},{-0.142,-0.99},{-0.959,0.282},{1.,0.},{-0.655,0.756},{0.655,0.756},{0.841,-0.541},{0.415,-0.91},{-0.841,-0.541}},{{0.959,0.282},{-0.142,-0.99},{-0.841,0.541},{0.841,-0.541},{-0.415,0.91},{0.415,-0.91},{-1.,0.},{-0.841,-0.541},{0.841,0.541},{-0.415,-0.91},{0.415,0.91},{0.959,-0.282},{-0.142,0.99},{0.142,-0.99},{-0.959,0.282},{-0.655,0.756},{1.,0.},{0.655,-0.756},{0.655,0.756},{-0.655,-0.756},{-0.959,-0.282},{0.142,0.99}},{{0.959,0.282},{-1.,0.},{-0.841,0.541},{-0.415,0.91},{0.655,0.756},{0.142,0.99},{-0.142,-0.99},{0.415,-0.91},{1.,0.},{-0.959,-0.282},{0.841,-0.541},{-0.655,-0.756},{0.841,0.541},{0.415,0.91},{-0.959,0.282},{-0.655,0.756},{-0.415,-0.91},{-0.142,0.99},{0.959,-0.282},{-0.841,-0.541},{0.142,-0.99},{0.655,-0.756}},{{0.959,0.282},{-0.841,-0.541},{-0.415,-0.91},{-0.841,0.541},{-1.,0.},{0.415,0.91},{-0.142,0.99},{0.841,-0.541},{0.841,0.541},{0.415,-0.91},{-0.142,-0.99},{-0.655,0.756},{1.,0.},{-0.959,-0.282},{-0.655,-0.756},{-0.959,0.282},{-0.415,0.91},{0.655,0.756},{0.142,-0.99},{0.655,-0.756},{0.142,0.99},{0.959,-0.282}},{{0.959,0.282},{0.415,-0.91},{0.142,0.99},{-0.415,0.91},{-1.,0.},{-0.841,0.541},{-0.142,-0.99},{-0.655,-0.756},{0.841,0.541},{0.655,-0.756},{0.415,0.91},{0.959,-0.282},{-0.959,-0.282},{-0.959,0.282},{0.142,-0.99},{-0.142,0.99},{1.,0.},{-0.655,0.756},{0.655,0.756},{0.841,-0.541},{-0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RawGraphData["CrossingNumberGraph7B","CochromaticGraphNames"]:={}
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RawGraphData["CrossingNumberGraph7B","ChromaticNumber"]:=3
RawGraphData["CrossingNumberGraph7B","EdgeConnectivity"]:=3
RawGraphData["CrossingNumberGraph7B","VertexConnectivity"]:=3
RawGraphData["CrossingNumberGraph7B","ChromaticPolynomial"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","SigmaPolynomial"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","EdgeChromaticNumber"]:=3
RawGraphData["CrossingNumberGraph7B","IndependenceNumber"]:=10
RawGraphData["CrossingNumberGraph7B","LovaszNumber"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","ShannonCapacity"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","Genus"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","CliqueNumber"]:=2
RawGraphData["CrossingNumberGraph7B","CyclomaticNumber"]:=12
RawGraphData["CrossingNumberGraph7B","WienerIndex"]:=588
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RawGraphData["CrossingNumberGraph7B","KirchhoffSumIndex"]:=2404928675/25781448
RawGraphData["CrossingNumberGraph7B","BalabanIndex"]:=(33(188/477+Sqrt[3/106]+1/Sqrt[330]))/13
RawGraphData["CrossingNumberGraph7B","CrossingNumber"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","RectilinearCrossingNumber"]:=7
RawGraphData["CrossingNumberGraph7B","ToroidalCrossingNumber"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","ProjectivePlaneCrossingNumber"]:=Missing["NotAvailable"]
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RawGraphData["CrossingNumberGraph7B","HamiltonianCycleCount"]:=56
RawGraphData["CrossingNumberGraph7B","Cycles"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","CycleCount"]:=Missing["NotAvailable"]
RawGraphData["CrossingNumberGraph7B","FaceIndices"]:=Missing["NotApplicable"]
RawGraphData["CrossingNumberGraph7B","FaceCount"]:=Missing["NotApplicable"]

...delete...

7B [was 7E]

GraphData["CrossingNumberGraph7B","AllImages"]

GraphPlot[Rule@@@Flatten[{Partition[Range[22],2,1,1],{{1,15},{2,8},{3,17},{4,21},{5,11},{6,18},{7,13},{9,20},{10,16},{12,22},{14,19}}},1],
VertexCoordinateRules-> Thread[Range[22]-> {{0,0},{2,0},{7,0},{7,7},{6,6},{6,5},{2,5},{2,3},{4,3},{5,3},{5,6},{1,6},{1,5},{1,2},{1,1},{5,1},{6,1},{6,2},{3,2},{3,4},{3,7},{0,7}}]]

g=//ToCombinatoricaGraph

CrossingNumber[g]

r=RecognizeGraph[g]

CrossingNumber[Graph[r]]

GraphData["CrossingNumberGraph7E","AllImages"]

ToCommonEdges[g,Graph[r]]//Round

GraphData["CrossingNumberGraph7E","AllImages"]

7C

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]//Round]

CrossingNumber[g]

7D

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

CrossingNumber[g]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]//Round]

8A == McGeeGraph

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

CrossingNumber[g]

CrossingNumberGraph8A =  Graph[{{{1, 4}}, {{1, 7}}, {{1, 23}}, {{2, 3}}, {{2, 12}}, {{2, 24}}, {{3, 5}}, {{3, 8}}, 
  {{4, 6}}, {{4, 12}}, {{5, 7}}, {{5, 10}}, {{6, 9}}, {{6, 11}}, {{7, 14}}, {{8, 9}}, 
  {{8, 13}}, {{9, 15}}, {{10, 11}}, {{10, 16}}, {{11, 18}}, {{12, 17}}, {{13, 19}}, 
  {{13, 23}}, {{14, 15}}, {{14, 21}}, {{15, 20}}, {{16, 17}}, {{16, 19}}, {{17, 20}}, 
  {{18, 22}}, {{18, 24}}, {{19, 21}}, {{20, 22}}, {{21, 24}}, {{22, 23}}}, 
{{{0, 0}}, {{0, 7}}, {{1, 3}}, {{1, 4}}, {{2, 3}}, {{2, 4}}, {{3, 1}}, {{3, 
   2}}, {{3, 3}}, {{3, 4}}, {{3, 5}}, {{3, 6}}, {{4, 1}}, {{4, 2}}, {{4, 
   3}}, {{4, 4}}, {{4, 5}}, {{4, 6}}, {{5, 3}}, {{5, 4}}, {{6, 3}}, {{6, 
   4}}, {{7, 0}}, {{7, 7}}}];

r=RecognizeGraph[g=CrossingNumberGraph8A]

CrossingNumber[g]

GraphPlot[g,Method->None,ImageSize->200]

ToCommonEdges[g,Graph[r]]

GraphData[r,"StandardNames"]

Insert[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]],6]

8B == Foster024A

r=RecognizeGraph[g1=ToCombinatoricaGraph[]]

GraphData[r,"StandardNames"]

CrossingNumber[g]

GraphData[r,"AllImages"]

ShowGraph[g2=Graph[{#}&/@{{1,2},{1,6},{1,7},{2,3},{2,17},{3,4},{3,19},{4,5},{4,11},{5,9},{5,10},{6,11},{6,22},{7,8},{7,12},{8,9},{8,23},{9,13},{10,12},{10,14},{11,15},{12,16},{13,15},{13,18},{14,19},{14,21},{15,20},{16,17},{16,20},{17,18},{18,24},{19,24},{20,21},{21,22},{22,23},{23,24}},{#}&/@Drop[Tuples[Range[5],{2}],{13}]]]

RecognizeGraph[g2]

CrossingNumber[g2]

Join[GraphData[r,"Embeddings"],{ToCommonEdges[g1,Graph[r]]//Round,ToCommonEdges[g2,Graph[r]]}]

8C [wrong # of crossings]

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

CrossingNumber[g]

GraphData[r,"AllImages"]

HamiltonianCycles[g1,All]//Length

file=OpenWrite["cn.m"]

WriteString[file,GraphDataString[g,
"StandardName"->"CrossingNumberGraph8C",
"Name"->"8-crossing number graph C",
Debug->True,
TimeConstraint->10]
]

Closed[file]

8C

r=RecognizeGraph[g=ToCombinatoricaGraph[]]

r=RecognizeGraph[g2=ToCombinatoricaGraph[]]

CrossingNumber[g2]

GraphData[r,"AllImages"]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g2,Graph[r]]//Round]

WriteString[file,GraphDataString[g1,
"StandardName"->"CrossingNumberGraph8C",
"Name"->"8-crossing number graph C",
"RectilinearCrossingNumber"->8,
Debug->True,
TimeConstraint->10]
]

8D [wrong; has CN=7]

Noted by E. Weisstein around 2019-04-01 that this graph has crossing number 7 (not 8 as claimed in the TMJ article).

r=RecognizeGraph[g1=ToCombinatoricaGraph[]]

r=RecognizeGraph[g2=ToCombinatoricaGraph[]]

r=RecognizeGraph[g3= ToCombinatoricaGraph[]]

CrossingNumber[g3]

GraphData[r,"AllImages"]

LCF[Graph[r],All]

Round[ToCommonEdges[g3,Graph[r]]]

Prepend[GraphData[r,"Embeddings"],ToCommonEdges[g3,Graph[r]]//Round]

GraphData["CrossingNumberGraph8D","Graph","All"]

RectilinearCrossingCount/@GraphData["CrossingNumberGraph8D","Graph","All"]

GraphData["CrossingNumberGraph8D",{"V","CN","RCN"}]

CrossingNumber[GraphData["CrossingNumberGraph8D"],MaxIterations->100]

8E [wrong; has CN=7]

Noted by E. Weisstein around 2019-04-01 that this graph has crossing number 7 (not 8 as claimed in the TMJ article).

r=RecognizeGraph[g= ToCombinatoricaGraph[]]

CrossingNumber[g]

GraphData[r,"AllImages"]

GraphData["CrossingNumberGraph8E","Graph","All"]

RectilinearCrossingCount/@GraphData["CrossingNumberGraph8E","Graph","All"]

GraphData["CrossingNumberGraph8E",{"V","CN","RCN"}]

CrossingNumber[GraphData["CrossingNumberGraph8E"],Method->3,MaxIterations->100]

9A (prior to V12, incorrectly called 10A)

Noted by E. Weisstein around 2019-04-01 that this graph has crossing number 9 (not 10 as claimed in the TMJ article).

10-crossing:

g1=System`Graph[Range[26],UndirectedEdge@@@{{1,5},{1,7},{1,25},{2,3},{2,6},{2,24},{3,4},{3,12},{4,8},{4,11},{5,6},{5,9},{6,15},{7,8},{7,22},{8,14},{9,10},{9,13},{10,11},{10,18},{11,21},{12,13},{12,16},{13,19},{14,15},{14,17},{15,20},{16,17},{16,25},{17,18},{18,23},{19,20},{19,22},{20,21},{21,26},{22,23},{23,24},{24,26},{25,26}},VertexCoordinates->{{0,0},{0,8},{2,2},{2,3},{2,4},{2,7},{4,1},{4,3},{4,4},{4,5},{4,6},{6,2},{6,3},{6,4},{6,6},{8,2},{8,4},{8,5},{10,3},{10,6},{10,7},{12,2},{12,5},{12,8},{14,0},{14,7}}]

RecognizeGraph[g1]

Construction via edge insertion:

mcgee=GraphData["McGeeGraph","Graph",{"Labeled","MinimalRectilinearCrossing"}][[1]]

GraphData["McGeeGraph","RCN"]

EdgeInsert[mcgee,#]&/@{{3\[UndirectedEdge]15,9\[UndirectedEdge]21},{6\[UndirectedEdge]18,12\[UndirectedEdge]24}}

RecognizeGraph/@%

RectilinearCrossingCount@g1

CrossingNumber[g1,MaxIterations->100]

ShowGraph[g,AspectRatio-> 1,ImageSize->200]

9-crossing:

g2=Graph[Range[26],UndirectedEdge@@@{{1,5},{1,7},{1,25},{2,3},{2,6},{2,24},{3,4},{3,12},{4,8},{4,11},{5,6},{5,9},{6,15},{7,8},{7,22},{8,14},{9,10},{9,13},{10,11},{10,18},{11,21},{12,13},{12,16},{13,19},{14,15},{14,17},{15,20},{16,17},{16,25},{17,18},{18,23},{19,20},{19,22},{20,21},{21,26},{22,23},{23,24},{24,26},{25,26}},VertexCoordinates->{{4,9},{2,2},{2,1},{1,1},{4,2},{3,2},{13,0},{3,0},{5,2},{5,3},{1,3},{3,1},{6,2},{8,1},{9,1},{5,1},{7,1},{7,3},{8,2},{10,1},{-1,-1},{8,4},{7,4},{2,4},{-3,-2},{1,4}}]

r=RecognizeGraph@g2

RectilinearCrossingCount@g2

v=ToCommonEdges[g2,GraphData[r]]

9B == excised Coxeter

From Ed, Apr 5, 2019

coxeter=GraphData["CoxeterGraph","Graph",{"Labeled","MinimalCrossing"}][[1]]

gs=EdgeExcise[coxeter,UndirectedEdge@@#]&/@{{16,17},{3,28},{9,10},{18,19}}

CubicGraphQ/@gs

VertexCount/@gs

IsomorphicGraphQ@@gs

RecognizeGraph/@gs

CrossingNumber[IndexGraph[#],MaxIterations->100]&/@gs//Timing

Rectilinear crossing embedding from Ed Pegg, 2019-04-08:

GraphPlot[g=Graph[Range[26],UndirectedEdge@@@{{1,2},{1,14},{1,22},{2,3},{2,8},{3,4},{3,26},{4,5},{4,25},{5,6},{5,24},{6,7},{6,16},{7,8},{7,23},{8,9},{9,10},{9,20},{10,11},{10,15},{11,12},{11,26},{12,13},{12,24},{13,14},{13,23},{14,15},{15,25},{16,17},{16,22},{17,18},{17,26},{18,19},{18,23},{19,20},{19,25},{20,21},{21,22},{21,24}},VertexCoordinates->{{0,0},{0,8},{1,7},{1,4},{2,2},{1,1},{8,0},{8,8},{7,7},{3,5},{4,5},{4,4},{4,3},{2,3},{2,4},{5,1},{5,4},{6,4},{6,6},{7,4},{6,2},{7,1},{6,3},{3,2},{2,6},{5,5}}],VertexLabeling->True]

r=RecognizeGraph[g]

RectilinearCrossingCount[g]

ToCommonEdges[g,GraphData[r]]

GetGraphData[r,"File"]

10A (28-node graph from McGee double insertion or Levi excision)

Not known if there is a 26-node graph with 10 crossings

VertexCount[g0=Graph[UndirectedEdge@@@{{1,2},{1,8},{1,24},{2,3},{2,19},{3,4},{4,5},{4,11},{5,6},{5,22},{6,7},{7,8},{7,14},{8,9},{9,10},{10,11},{10,17},{11,12},{12,13},{13,14},{13,20},{14,15},{15,16},{16,17},{16,23},{17,18},{18,19},{19,20},{20,21},{21,22},{22,23},{23,24},{3,25},{15,25},{9,26},{21,26},{25,26},{6,27},{18,27},{12,28},{24,28},{27,28}}]]

RecognizeGraph[g]

From McGee edge insertion

mcgee=GraphData["McGeeGraph","Graph",{"Labeled","MinimalRectilinearCrossing"}][[1]]

cng10a=EdgeInsert[mcgee,{{3\[UndirectedEdge]15,9\[UndirectedEdge]21},{6\[UndirectedEdge]18,12\[UndirectedEdge]24}}]

IsomorphicGraphQ[g0,cng10a]

CrossingNumber[g,MaxIterations->100]//Timing

CrossingNumber[g,Method->1,MaxIterations->10]//Timing

CrossingNumber[g,Method->1,MaxIterations->100]//Timing

From Levi edge excision

g1=EdgeExcise[GraphData["LeviGraph","Graph","MinimalCrossing"][[1]],13\[UndirectedEdge]26]

CubicGraphQ[g1]

VertexCount[g1]

r=RecognizeGraph[g1]

RectilinearCrossingCount@g1

ToCommonEdges[g1,GraphData[r]]

Rectilinear crossing from Ed Pegg 2019-04-08:

GraphPlot[g2=Graph[Range[28],UndirectedEdge@@@{{1,2},{1,13},{1,28},{2,3},{2,18},{3,4},{3,23},{4,5},{4,11},{5,6},{5,26},{6,7},{6,14},{7,8},{7,19},{8,9},{8,24},{9,10},{9,28},{10,11},{10,16},{11,12},{12,20},{12,13},{13,14},{14,15},{15,16},{15,22},{16,17},{17,18},{17,25},{18,19},{19,20},{20,21},{21,22},{21,27},{22,23},{23,24},{24,25},{25,26},{26,27},{27,28}},VertexCoordinates->{{0,0},{0,9},{1,7},{1,1},{2,2},{4,2},{4,3},{4,5},{3,5},{3,4},{3,1},{8,1},{8,0},{6,2},{6,3},{5,3},{5,4},{7,7},{7,2},{8,8},{2,8},{6,7},{4,7},{4,6},{5,5},{3,6},{2,6},{2,5}}],VertexLabeling->True]

RectilinearCrossingCount[g2]

r=RecognizeGraph[g2]

GetGraphData[r,"File"]

ToCommonEdges[g2,GraphData[r]]

file=OpenWrite["~/cng10a.m"]

WriteString[file,GraphDataString[cng10a,
"StandardName"->"CrossingNumberGraph10A",
"Name"->"10\[Hyphen]crossing number graph A",
"CrossingNumber"->10,
"RectilinearCrossingNumber"->10,
"Information"->"SmallestCubicCrossingNumberGraph",
TimeConstraint->10,
Debug->True]]

Close[file]

10B (28-node graph)

Clancy et al. 2019 preprint

Length[v={{1042.3741804587553`,588.3977186422068`},{970.4702147245027`,537.5840950018468`},{970.4702147245027`,588.529596683198`},{912.6670549116118`,530.1887802416506`},{921.6246172343183`,588.7223415123389`},{981.7305705322087`,715.5890169458269`},{722.2655971144634`,656.5177990464824`},{723.6959666359828`,713.326801319594`},{730.5434803028311`,763.6534906547558`},{779.1760440344912`,766.4939407684114`},{779.1760440344912`,817.6220428142118`},{668.5506565722981`,761.2898303816067`},{671.2998065037291`,401.87144635725997`},{726.4045387086472`,404.7118964709156`},{722.1742969322388`,455.80956512264106`},{781.6411489545566`,452.9488260796022`},{774.1646784768275`,405.4828757874792`},{913.1438447521182`,402.62213674444035`},{971.3933610114407`,402.62213674444035`},{969.9832804193045`,344.70738781994146`},{1030.05880032312`,344.70738781994146`},{1034.7252540812685`,184.8914910680487`},{1034.7252540812685`,248.41612896701747`},{725.6842817155417`,242.07583853475057`},{670.0013150232007`,241.67005994708552`},{602.6420694707971`,240.2396904255661`},{606.9331780353554`,174.2296586771488`},{611.2242865999136`,119.71330542434498`}}/1042.374]

cng10b=Graph[Range[Length[v]],UndirectedEdge@@@Sort[Sort/@{{28,27},{27,26},{26,25},{25,24},{24,23},{22,23},{28,22},{25,13},{13,14},{14,15},{15,16},{16,17},{17,18},{18,19},{19,20},{20,21},{21,2},{2,4},{4,5},{5,6},{1,19},{22,1},{23,3},{3,10},{6,11},{10,11},{10,9},{9,12},{9,8},{8,7},{28,12},{7,5},{7,26},{11,13},{1,6},{12,15},{17,24},{14,20},{2,3},{4,16},{8,18},{21,27}}],VertexCoordinates->v,VertexLabels->"Name"]

VertexDegree[cng10b]

CubicGraphQ[cng10b]

CrossingNumber[cng10b,MaxIterations->200]//Timing

RectilinearCrossingCount[cng10b]

RecognizeGraph[cng10b]

Rectilinear crossing version from Ed 2019-04-08:

GraphPlot[g=Graph[{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28},{Null,SparseArray[Automatic,{28,28},0,{1,{{0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84},{{6},{19},{22},{3},{4},{21},{2},{10},{23},{2},{5},{16},{4},{6},{7},{1},{5},{11},{5},{8},{26},{7},{9},{18},{8},{10},{12},{3},{9},{11},{6},{10},{13},{9},{15},{28},{11},{14},{25},{13},{15},{20},{12},{14},{16},{4},{15},{17},{16},{18},{24},{8},{17},{19},{1},{18},{20},{14},{19},{21},{2},{20},{27},{1},{23},{28},{3},{22},{24},{17},{23},{25},{13},{24},{26},{7},{25},{27},{21},{26},{28},{12},{22},{27}}},Pattern}]},{VertexCoordinates->{{7,1},{5,3},{6,3},{5,4},{5,5},{7,8},{2,6},{4,6},{4,7},{6,7},{0,8},{1,7},{2,3},{2,4},{2,5},{3,5},{3,4},{4,4},{4,3},{4,2},{5,2},{6,0},{5,1},{3,2},{2,2},{1,4},{1,1},{0,0}}}],VertexLabeling->True]

r=RecognizeGraph[g]

RectilinearCrossingCount@g

ToCommonEdges[g,GraphData[r]]

file=OpenWrite["~/cng10b.m"]

WriteString[file,GraphDataString[cng10b,
"StandardName"->"CrossingNumberGraph10B",
"Name"->"10\[Hyphen]crossing number graph B",
"CrossingNumber"->10,
"RectilinearCrossingNumber"->10,
"Information"->"SmallestCubicCrossingNumberGraph",
TimeConstraint->10,
Debug->True]]

Close[file]

11A == CoxeterGraph

CrossingNumberGraph11A =Graph[{{{1, 2}}, {{1, 8}}, {{1, 24}}, {{2, 5}}, {{2, 28}}, {{3, 6}}, {{3, 7}}, {{3, 27}}, 
  {{4, 5}}, {{4, 7}}, {{4, 10}}, {{5, 18}}, {{6, 12}}, {{6, 19}}, {{7, 11}}, {{8, 9}}, 
  {{8, 14}}, {{9, 10}}, {{9, 12}}, {{10, 23}}, {{11, 15}}, {{11, 20}}, {{12, 13}}, 
  {{13, 16}}, {{13, 26}}, {{14, 15}}, {{14, 21}}, {{15, 16}}, {{16, 18}}, {{17, 18}}, 
  {{17, 19}}, {{17, 22}}, {{19, 24}}, {{20, 24}}, {{20, 25}}, {{21, 22}}, {{21, 27}}, 
  {{22, 23}}, {{23, 25}}, {{25, 26}}, {{26, 28}}, {{27, 28}}}, 
 {{{0,0}},{{0,8}},{{1,1}},{{1,4}},{{1,7}},{{2,1}},{{2,2}},{{2,3}},{{2,4}},{{2,6}},{{3,2}},{{3,4}},{{3,5}},{{4,3}},{{4,4}},{{4,5}},{{5,4}},{{5,5}},{{6,1}},{{6,2}},{{6,3}},{{6,4}},{{6,6}},{{7,1}},{{7,4}},{{7,7}},{{8,0}},{{8,8}}}];

r=RecognizeGraph[g=CrossingNumberGraph11A]

CrossingNumber[g]

ShowGraph[g,AspectRatio-> 1,ImageSize->200]

GraphData[r,"AllImages"]

Append[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]]

12A

from Michael Haythorpe:
Forwarded by Ed 2019-04-12

For 30-vertex cubic graphs, of course there is the Levi graph with girth 8, it has crossing number 13. Then there are 545 graphs with girth 7. I looked and all but one of them has crossing number less than 12.

However, there was one graph for which the best QuickCross could do was 12 crossings. I am willing to bet it has crossing number 12, although I have not checked it exhaustively yet. I'm not sure if it is famous graph (or a modification of one) or not. The edgelist is below, do you recognise it?

gg=Graph[UndirectedEdge@@@{1->2,1->3,1->4,2->5,2->6,3->7,3->8,4->9,4->10,5->11,5->12,6->13,6->14,7->15,7->16,8->17,8->18,9->19,9->20,10->21,10->22,11->15,11->19,12->17,12->21,13->23,13->24,14->25,14->26,15->27,16->23,16->28,17->29,18->24,18->30,19->30,20->23,20->29,21->28,22->24,22->27,25->27,25->29,26->28,26->30}]

RecognizeGraph[gg]

Through[{VertexCount,CubicGraphQ}[gg]]

GraphData["Cubic",30]

CrossingNumber[gg,Method->1,MaxIterations->500]//Timing

CrossingNumber[gg,MaxIterations->500]//Timing

ExportString[gg,"Graph6"]

./QC -g ']sP@@?OC?O@?@?@??P?C??P?@??@O?@???S??C???Ca??Oa??O???O???CAO??`A??AO_??E@?' -r 1 -z 200
Current best soln: 12, at seed: 35, current seed: 10000

./QC -g ']sP@@?OC?O@?@?@??P?C??P?@??@O?@???S??C???Ca??Oa??O???O???CAO??`A??AO_??E@?' -r 1 -z 200 -e 1
Current best soln: 12, at seed: 10, current seed: 1000

./QC -g ']sP@@?OC?O@?@?@??P?C??P?@??@O?@???S??C???Ca??Oa??O???O???CAO??`A??AO_??E@?' -r 1 -z 200 -e 2
Current best soln: 12, at seed: 33, current seed: 10000

Ed claimed this doesn't give a known graph upon edge excision, but it actually does:

DeleteCases[{#,RecognizeGraph[EdgeExcise[gg,#]]}&/@EdgeList[gg],{_,{}}]

List@@@%[[All,1]]

Construct it as an insertion:

Length[s=Subsets[EdgeList[GraphData["CrossingNumberGraph10A"]],{2}]]

inserts=Cases[{#,IsomorphicGraphQ[gg,EdgeInsert[GraphData["CrossingNumberGraph10A"],#]]}&/@s,{e_,True}:>e]

EdgeInsert[GraphData["CrossingNumberGraph10A","Graph","MinimalCrossing"][[1]],#]&/@inserts

{#,RectilinearCrossingCount@EdgeInsert[GraphData["CrossingNumberGraph10A","Graph","MinimalCrossing"][[1]],#]}&/@inserts

Embedding from Ed 2019-04-11:

g=;

r=RecognizeGraph[g]

RectilinearCrossingCount@g

ToCommonEdges[g,GraphData[r]]

GetGraphData[r,"File"]

file=OpenWrite["~/cng12a.m"]

GraphData["CrossingNumberGraph10A","Name"]

WriteString[file,GraphDataString[gg,
"StandardName"->"CrossingNumberGraph12A",
"AlternateStandardNames"->{"CNG12A"},
"Name"->"12\[Hyphen]crossing number graph A",
"Information"->"SmallestCubicCrossingNumberGraph",
"CrossingNumber"->12,
"RectilinearCrossingNumber"->12,
Debug->True,
TimeConstraint->10]]

Close[file]

13A == LeviGraph

CrossingNumberGraph13A =Graph[
{{{1, 2}}, {{1, 6}}, {{1, 29}}, {{2, 4}}, {{2, 26}}, {{3, 4}}, {{3, 5}}, {{3, 9}},   {{4, 17}}, {{5, 12}}, {{5, 13}}, {{6, 7}}, {{6, 11}}, {{7, 8}}, {{7, 12}}, {{8, 24}},   {{8, 28}}, {{9, 10}}, {{9, 27}}, {{10, 11}}, {{10, 18}}, {{11, 15}}, {{12, 20}},   {{13, 14}}, {{13, 22}}, {{14, 15}}, {{14, 25}}, {{15, 16}}, {{16, 17}}, {{16, 21}},   {{17, 24}}, {{18, 19}}, {{18, 23}}, {{19, 20}}, {{19, 26}}, {{20, 21}}, {{21, 30}},   {{22, 23}}, {{22, 29}}, {{23, 24}}, {{25, 26}}, {{25, 28}}, {{27, 28}}, {{27, 30}},   {{29, 30}}}, {{{0,0}},{{0,9}},{{1,1}},{{1,7}},{{2,2}},{{2,5}},{{2,6}},{{2,8}},{{3,1}},{{3,4}},{{3,5}},{{3,6}},{{4,2}},{{4,3}},{{4,5}},{{4,6}},{{4,7}},{{5,3}},{{5,4}},{{5,5}},{{5,6}},{{6,2}},{{6,3}},{{6,7}},{{7,2}},{{7,7}},{{8,1}},{{8,8}},{{9,0}},{{9,9}}}];

r=RecognizeGraph[g=CrossingNumberGraph13A]

CrossingNumber[g]

ShowGraph[g,AspectRatio-> 1,ImageSize->200]

Append[GraphData[r,"Embeddings"],ToCommonEdges[g,Graph[r]]]

GraphData[r,"AllImages"]

Embeddings

<<MathWorld`Graphs`

SortBy[GraphData["CrossingNumberGraph*"],GraphData[#,"V"]&]

"CrossingNumberGraph2B"

Names

GraphData["CrossingNumberGraph2B","StandardName"]

GraphData["CrossingNumberGraph2B","StandardNames"]

Construction from Mohar and Thomassen 2001

Using the numbering of Glover et al. 1979 for projective planar forbidden topological minors

Primary

GraphData["CrossingNumberGraph2B"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph2B","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph2B","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph2B","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph2B","Edges"]

v=Join[UnitCirclePoints[8],{#/2,0}&/@{-1,1}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[8,6,5,7,8,1,2,4,3,10,2],e[7,9,6],{1,3},{4,5},{9,10}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

v={{0,0},{1,0},{1,1},{0,1},{1/3,1/4},{2/3,1/4},{2/3,3/4},{1/3,3/4},{1/3,1/2},{2/3,1/2}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,7,10,6,3,4,5,9,8,1,5],{4,8},{3,7},{2,6},{9,10}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph2B","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph2B","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph2B","Graph","Circular"]//StyleGraphs

Circular construction

Length[hc=FindHamiltonianCycle[GraphData["CrossingNumberGraph2B"],All][[All,All,1]]]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData["CrossingNumberGraph2B"],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Quiet//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]//Quiet

quad

Planar

GraphData["CrossingNumberGraph2B","Planar"]

GraphData["CrossingNumberGraph2B","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph2B",{"RCN","CN"}]

GraphData["CrossingNumberGraph2B","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph2B"],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph2B"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph2B","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossingNumberGraph2B"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

GraphPlot@

GraphPlot@RotateGraph[,Pi/5]

v={{0,0},{1,0},{1,1},{0,1},{1/5,1/3},{2/5,1/3},{3/5,1/3},{4/5,1/3},{1/5,2/3},{4/5,2/3}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,7,8,2,6,5,1,4,3,2],e[5,9,4],e[8,10,3],{6,10},{7,9}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph2B","Circulant"]

GraphData["CrossingNumberGraph2B","Graph","Circulant"]//StyleGraphs

Line graph

GraphData["CrossingNumberGraph2B","Line"]

LineGraphQ[GraphData["CrossingNumberGraph2B"]]//Timing

Select[GraphData[All],GraphData[#,"LineGraph","Name"]==="CrossingNumberGraph2B"&]

LineGraphRootGraph[GraphData["CrossingNumberGraph2B"]]

Halin

GraphData["CrossingNumberGraph2B","Halin"]

GraphData["CrossingNumberGraph2B","Graph","Halin"]//StyleGraphs

Rigid

GraphData["CrossingNumberGraph2B","Rigid"]

GraphData["CrossingNumberGraph2B","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["CrossingNumberGraph2B","Flexible"]

GraphData["CrossingNumberGraph2B","Graph","Flexible"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph2B","LCF"]

GraphData["CrossingNumberGraph2B","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph2B","LCFSignature"]

GraphData["CrossingNumberGraph2B","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph2B"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph2B"],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

RotateGraph[,Pi/2]

UnitDistance

GraphData["CrossingNumberGraph2B","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph2B"],Debug->True]//Timing

GraphData["CrossingNumberGraph2B","Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph2B","VertexList"],GraphData["CrossingNumberGraph2B","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData["CrossingNumberGraph2B","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph2B"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData["CrossingNumberGraph2B","Matchstick"]

GraphData["CrossingNumberGraph2B","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph2B","Edges"]]

3D All

GraphData["CrossingNumberGraph2B","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph2B","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph2B","Polyhedral"]

GraphData["CrossingNumberGraph2B","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph3A"

GraphData["CrossingNumberGraph3A","StandardName"]

Primary

GraphData["CrossingNumberGraph3A"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph3A","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph3A","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph3A","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph3A","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph3A","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph3A","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph3A","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph3A","Planar"]

GraphData["CrossingNumberGraph3A","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph3A",{"RCN","CN"}]

GraphData["CrossingNumberGraph3A","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph3A"],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph3A"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph3A","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossingNumberGraph3A"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

GraphPlot@

v={{-1,-1},{0,-1},{1,-1},{2,0},{1,1},{0,1},{-1,1},{-2,0},{-3/2,0},{-1,-1/2},{0,1/2},{1,1/2},{3/2,0},{1/2,0}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,8,1,10,7],e[8,9,14,13,4],e[2,11,6],e[3,12,5],{9,10},{11,14},{12,13}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph3A","Circulant"]

GraphData["CrossingNumberGraph3A","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph3A","Halin"]

GraphData["CrossingNumberGraph3A","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph3A","LCF"]

GraphData["CrossingNumberGraph3A","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph3A","LCFSignature"]

GraphData["CrossingNumberGraph3A","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph3A"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph3A"],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

UnitDistance

GraphData["CrossingNumberGraph3A","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph3A"],Debug->True]//Timing

GraphData["CrossingNumberGraph3A","Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph3A","VertexList"],GraphData["CrossingNumberGraph3A","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData["CrossingNumberGraph3A","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph3A"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData["CrossingNumberGraph3A","Matchstick"]

GraphData["CrossingNumberGraph3A","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph3A","Edges"]]

3D All

GraphData["CrossingNumberGraph3A","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph3A","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph3A","Polyhedral"]

GraphData["CrossingNumberGraph3A","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph3B"

GraphData["CrossingNumberGraph3B","StandardName"]

Primary

GraphData["CrossingNumberGraph3B"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph3B","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph3B","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph3B","EdgeList"]]

Default embedding

GraphPlot[GraphData["CrossingNumberGraph3B","Edges"],GraphLayout->"SpringEmbedding"]

v=Join[UnitCirclePoints[8]];
a=1/3;
v=Join[v,v[[#]]+{a,0}&/@{6,7},v[[#]]-{a,0}&/@{2,3},v[[#]]-{0,a}&/@{4,5}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,8,1,11,12,13,14,9,10,8],{7,9},{6,10},{5,13},{4,14},{2,12},{3,11}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph3B","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph3B","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph3B","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph3B","Planar"]

GraphData["CrossingNumberGraph3B","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph3B",{"RCN","CN"}]

GraphData["CrossingNumberGraph3B","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph3B"],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph3B"],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph3B","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossingNumberGraph3B"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph3B","Circulant"]

GraphData["CrossingNumberGraph3B","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph3B","Halin"]

GraphData["CrossingNumberGraph3B","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph3B","LCF"]

GraphData["CrossingNumberGraph3B","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph3B","LCFSignature"]

GraphData["CrossingNumberGraph3B","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph3B"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph3B"],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

UnitDistance

GraphData["CrossingNumberGraph3B","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph3B"],Debug->True]//Timing

GraphData["CrossingNumberGraph3B","Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph3B","VertexList"],GraphData["CrossingNumberGraph3B","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData["CrossingNumberGraph3B","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph3B"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData["CrossingNumberGraph3B","Matchstick"]

GraphData["CrossingNumberGraph3B","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph3B","Edges"]]

3D All

GraphData["CrossingNumberGraph3B","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph3B","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph3B","Polyhedral"]

GraphData["CrossingNumberGraph3B","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph3D"

GraphData["CrossingNumberGraph3D","StandardName"]

Primary

GraphData["CrossingNumberGraph3D"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph3D","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph3D","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph3D","EdgeList"]]

Pentagram construction

v[[14,1]]/v[[9,1]]//FullSimplify

v=Join[CirclePoints[9],CirclePoints[(Sqrt[2+(2/(Sqrt[5]))]) Sin[(π/9)],5]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,8,9,1,10,12,14,9],{4,8},{2,6},e[7,13,11,3],e[10,13],{5,12},{11,14}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph3D","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph3D","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph3D","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph3D","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph3D","Planar"]

GraphData["CrossingNumberGraph3D","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph3D",{"RCN","CN"}]

GraphData["CrossingNumberGraph3D","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph3D"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph3D"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph3D"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph3D"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph3D","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph3D","Circulant"]

GraphData["CrossingNumberGraph3D","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph3D","Halin"]

GraphData["CrossingNumberGraph3D","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph3D","LCF"]

GraphData["CrossingNumberGraph3D","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph3D","LCFSignature"]

GraphData["CrossingNumberGraph3D","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph3D"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph3D"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph3D","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph3D"],Debug->True]//Timing

GraphData["CrossingNumberGraph3D","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph3D","VertexList"],GraphData["CrossingNumberGraph3D","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph3D","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph3D"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph3D"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph3D"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"CrossingNumberGraph3D",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph3D"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,};

Matchstick

GraphData["CrossingNumberGraph3D","Matchstick"]

GraphData["CrossingNumberGraph3D","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph3D","Edges"]]

3D All

GraphData["CrossingNumberGraph3D","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph3D","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph3D","Polyhedral"]

GraphData["CrossingNumberGraph3D","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph3E"

GraphData["CrossingNumberGraph3E","StandardName"]

Primary

GraphData["CrossingNumberGraph3E"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph3E","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph3E","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph3E","EdgeList"]]

Default embedding

GraphPlot[GraphData["CrossingNumberGraph3E","Edges"],GraphLayout->"SpringEmbedding"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph3E","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph3E","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph3E","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph3E","Planar"]

GraphData["CrossingNumberGraph3E","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph3E",{"RCN","CN"}]

GraphData["CrossingNumberGraph3E","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph3E"],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph3E"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph3E","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossingNumberGraph3E"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph3E","Circulant"]

GraphData["CrossingNumberGraph3E","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph3E","Halin"]

GraphData["CrossingNumberGraph3E","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph3E","LCF"]

GraphData["CrossingNumberGraph3E","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph3E","LCFSignature"]

GraphData["CrossingNumberGraph3E","Graph","LCF"]//StyleGraphs

UnitDistance

GraphData["CrossingNumberGraph3E","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph3E"],Debug->True]//Timing

GraphData["CrossingNumberGraph3E","Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph3E","VertexList"],GraphData["CrossingNumberGraph3E","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData["CrossingNumberGraph3E","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph3E"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData["CrossingNumberGraph3E","Matchstick"]

GraphData["CrossingNumberGraph3E","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph3E","Edges"]]

3D All

GraphData["CrossingNumberGraph3E","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph3E","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph3E","Polyhedral"]

GraphData["CrossingNumberGraph3E","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph3F"

GraphData["CrossingNumberGraph3F","UnitDistance"]

GraphData["CrossingNumberGraph3F","Graph","All"]

GraphData["CrossingNumberGraph3F","Graph","UnitDistance"]

GraphData["CrossingNumberGraph3F",{"TriangleCount","SquareCount"}]

GraphPlot[GraphData["CrossingNumberGraph3F","EdgeList"]]

v=CirclePoints[Csc[Pi/8]/2,8];

vv=Join[v,{
v[[1]]+{0,sc},v[[8]]+{0,sc},
v[[2]]-{sc,0},v[[3]]-{sc,0},
v[[7]]+{sc,0},v[[6]]+{sc,0}
}]

Solve[FullSimplify[Cross[Append[vv[[4]]-vv[[13]],0],Append[vv[[12]]-vv[[10]],0]]]==0,sc]

N[%]

Graph[Range[Length[vv]],UndirectedEdge@@@Join[Partition[Range[8],2,1,1],{{8,10},{9,10},{1,9},{2,11},{11,12},{3,12},{7,13},{13,14},{6,14},{9,14},{10,12},{4,13},{5,11}}],VertexCoordinates->(vv/.sc->#),VertexLabels->"Name"]&/@{1/5,1/2,1,0,(1/2) (1+Cot[(π/8)])}

g=Graph[Range[Length[vv]],UndirectedEdge@@@Join[Partition[Range[8],2,1,1],{{8,10},{9,10},{1,9},{2,11},{11,12},{3,12},{7,13},{13,14},{6,14},{9,14},{10,12},{4,13},{5,11}}],VertexCoordinates->(vv/.sc->1/4),VertexLabels->"Name"]

<<MathWorld`Graphs`

RecognizeGraph[g]

All degenerate:

GraphPlot[Graph[Range[Length[vv]],UndirectedEdge@@@Join[Partition[Range[8],2,1,1],{{8,10},{9,10},{1,9},{2,11},{11,12},{3,12},{7,13},{13,14},{6,14},{9,14},{10,12},{4,13},{5,11}}],VertexCoordinates->(vv/.sc->1)]]

soln=Solve[{
(a-x)^2+(b-y)^2==1,
(x+1/2)^2+y^2==1,
(a+1/2)^2+(b-z)^2==1,
(x+1/2)^2+(y-z)^2==1,
(a+1/2)^2+(b-1)^2==1},{x,y,z,a,b}]

gs=Graph[Range[14],UndirectedEdge@@@{{1,2},{1,3},{1,10},{2,6},{2,9},{3,4},{3,11},{4,8},{4,9},{5,6},{5,7},{5,10},{6,12},{7,8},{7,13},{8,14},{9,11},{10,12},{11,13},{12,14},{13,14}},VertexCoordinates->({{-1/2,z},{1/2,z},{-a-1,b},{-a,b},{a,b},{a+1,b},{-1/2,1},{1/2,1},{-x,y},{x,y},{-x-1,y},{x+1,y},{-1/2,0},{1/2,0}}/.#),VertexLabels->"Name"]&/@soln

<<MathWorld`Graphs`

RecognizeGraph/@gs

UnitDistanceGraphEmbeddingQ/@gs

DegenerateEmbeddingQ/@gs

soln=Solve[{
(x+c-a+1/2)^2+(y+d-b-1)^2==1,
(x+1/2)^2+(y-z)^2==1,
(c-1/2)^2+(d-z)^2==1,
(x-a)^2+(y-b)^2==1,
(c-a)^2+(d-b)^2==1,
(a-1/2)^2+b^2==1,
b==1/2},{x,y,z,a,b,c,d},Reals]//RootReduce

gs=Graph[Range[14],UndirectedEdge@@@{{1,3},{1,11},{1,2},{2,10},{2,6},{3,4},{3,9},{4,10},{4,8},{5,7},{5,6},{5,11},{6,12},{7,8},{7,13},{8,14},{9,10},{9,13},{11,12},{12,14},{13,14}},VertexCoordinates->({{-1/2,z},{1/2,z},{-c,d},{-x-c+a,y+d-b},{x+c-a,y+d-b},{c,d},{-1/2,1},{1/2,1},{-a,b},{-x,y},{x,y},{a,b},{-1/2,0},{1/2,0}}/.#),VertexLabels->"Name"]&/@soln

RecognizeGraph[gs[[1]]]

EdgeLengths[gs[[1]]]//N

gs=Select[gs,UnitDistanceGraphEmbeddingQ[#]&&!DegenerateEmbeddingQ[#]&]

"CrossingNumberGraph3H"

GraphData["CrossingNumberGraph3H",{"TriangleCount","SquareCount"}]

GraphData["CrossingNumberGraph3H","Graph","MinimalCrossing"]

GraphData["CrossingNumberGraph3H","Graph","LCF"]

GraphData["CrossingNumberGraph3H","Graph","UnitDistance"]

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

GraphData["CrossingNumberGraph3H","Graph","All"]

GraphPlot[Graph[GraphData["CrossingNumberGraph3H","EdgeList"]]]

GraphData["CrossingNumberGraph3H","Graphics"]

soln=NSolve[{
a^2+(z+1-b)^2==1,
e^2+(h+1-f)^2==1,
c^2+(d-z)^2==1,
(a-e)^2+(b-f)^2==1,
(c-e)^2+(f-d)^2==1,
(c-x)^2+(d-y)^2==1,
x^2+(h-y)^2==1,
(a-1/2)^2+b^2==1,
(x+1/2)^2+y^2==1,
b==1/2},{h,z,a,b,c,d,e,f,x,y},Reals]

gs=Graph[Range[14],UndirectedEdge@@@{{1,4},{1,8},{1,9},{2,6},{2,5},{2,8},{3,6},{3,9},{3,7},{4,5},{4,7},{5,10},{6,11},{7,12},{8,13},{9,14},{10,11},{10,14},{11,12},{12,13},{13,14}},VertexCoordinates->({{0,z+1},{-e,f},{e,f},{0,z},{-c,d},{0,h+1},{c,d},{-a,b},{a,b},{-x,y},{0,h},{x,y},{-1/2,0},{1/2,0}}/.#),VertexLabels->"Name"]&/@soln

RecognizeGraph@gs[[1]]

Select[gs,UnitDistanceGraphEmbeddingQ[#]&&!DegenerateEmbeddingQ[#]&]

Do[
soln=NSolve[{
a^2+(z+1-b)^2==1,
e^2+(h+1-f)^2==1,
c^2+(d-z)^2==1,
(a-e)^2+(b-f)^2==1,
(c-e)^2+(f-d)^2==1,
(c-x)^2+(d-y)^2==1,
x^2+(h-y)^2==1,
(a-1/2)^2+b^2==1,
(x+1/2)^2+y^2==1},{h,z,a,c,d,e,f,x,y},Reals];
gs=Graph[Range[14],UndirectedEdge@@@{{1,4},{1,8},{1,9},{2,6},{2,5},{2,8},{3,6},{3,9},{3,7},{4,5},{4,7},{5,10},{6,11},{7,12},{8,13},{9,14},{10,11},{10,14},{11,12},{12,13},{13,14}},VertexCoordinates->({{0,z+1},{-e,f},{e,f},{0,z},{-c,d},{0,h+1},{c,d},{-a,b},{a,b},{-x,y},{0,h},{x,y},{-1/2,0},{1/2,0}}/.#),VertexLabels->"Name"]&/@soln;
Print[b->DeleteDuplicatesBy[Select[gs,UnitDistanceGraphEmbeddingQ[#]&&!DegenerateEmbeddingQ[#]&],Sort[Round[10^8GraphEmbedding[#]]]&]],
{b,-1,1,.1}]

gs={,,,,,,,,};

FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph3H"],"Results"->All]//Timing

gs={,}

"CrossingNumberGraph5B"

GraphData["CrossingNumberGraph5B","StandardName"]

Primary

GraphData["CrossingNumberGraph5B"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph5B","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph5B","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph5B","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph5B","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph5B","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph5B","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph5B","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph5B","Planar"]

GraphData["CrossingNumberGraph5B","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph5B",{"RCN","CN"}]

GraphData["CrossingNumberGraph5B","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph5B"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph5B"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph5B"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph5B"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph5B","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph5B","Circulant"]

GraphData["CrossingNumberGraph5B","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph5B","Halin"]

GraphData["CrossingNumberGraph5B","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph5B","LCF"]

GraphData["CrossingNumberGraph5B","LCFSignature"]

GraphData["CrossingNumberGraph5B","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph5B"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph5B"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph5B","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph5B"],Debug->True]//Timing

GraphData["CrossingNumberGraph5B","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph5B","VertexList"],GraphData["CrossingNumberGraph5B","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph5B","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph5B"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph5B"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph5B"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding original

{"CrossingNumberGraph5B",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph5B"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,};

UnitDistance by FindSubgraphUnitDistanceEmbedding new

{"CrossingNumberGraph5B",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph5B"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["CrossingNumberGraph5B","Matchstick"]

GraphData["CrossingNumberGraph5B","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph5B","Edges"]]

3D All

GraphData["CrossingNumberGraph5B","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph5B","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph5B","Polyhedral"]

GraphData["CrossingNumberGraph5B","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph6B"

Names

GraphData["CrossingNumberGraph6B","StandardName"]

GraphData["CrossingNumberGraph6B","StandardNames"]

Primary

GraphData["CrossingNumberGraph6B"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph6B","Graph","All"]//StyleGraphs

Configuration

GraphData["CrossingNumberGraph6B","Configuration"]

GraphData["CrossingNumberGraph6B","Graph","Configuration"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph6B","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph6B","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph6B","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

g=LabelGraph@

GraphPlot[GraphData["CrossingNumberGraph6B","EdgeList"]]

v=Join[CirclePoints[1/4,6],CirclePoints[1,10]];
v=Join[v,(v[[#]]-{0,1/2})&/@{12,11},(v[[#]]+{0,1/2})&/@{16,7}];
g=Graph[Range[Length[v]],UndirectedEdge@@@Join[Partition[Range[7,16],2,1],Partition[{16,6,2,4,19,15},2,1],Partition[{19,7,1,18},2,1],{{13,17},{17,18},{18,10},{12,4},{3,11},{14,5},{2,9},{16,20},{3,20},{5,3},{6,17},{1,5},{8,20}}],VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph6B","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["CrossingNumberGraph6B","Graph","Degenerate"]

Perspective

GraphData["CrossingNumberGraph6B","Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData["CrossingNumberGraph6B","Polyhedron"]

GeneralizedPetersen

GraphData["CrossingNumberGraph6B","Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData["CrossingNumberGraph6B","Graph","IGraph"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph6B","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph6B","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph6B","Planar"]

GraphData["CrossingNumberGraph6B","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph6B",{"RCN","CN"}]

GraphData["CrossingNumberGraph6B","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph6B"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CrossingNumberGraph6B"],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CrossingNumberGraph6B"],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CrossingNumberGraph6B"],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CrossingNumberGraph6B"],TimeConstraint->5 3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["CrossingNumberGraph6B"],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph6B"],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph6B","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossingNumberGraph6B"],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union//Timing

GraphPlot@RotateGraph[,-Pi/6]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphPlot@RotateGraph[,Pi/6]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph6B","Circulant"]

GraphData["CrossingNumberGraph6B","Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData["CrossingNumberGraph6B","ComplementGraph","Name"]

Line graph

GraphData["CrossingNumberGraph6B","Line"]

LineGraphQ[GraphData["CrossingNumberGraph6B"]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "CrossingNumberGraph6B"&]

RootGraph[GraphData["CrossingNumberGraph6B"]]

Halin

GraphData["CrossingNumberGraph6B","Halin"]

GraphData["CrossingNumberGraph6B","Graph","Halin"]//StyleGraphs

Rigid

GraphData["CrossingNumberGraph6B","Rigid"]

GraphData["CrossingNumberGraph6B","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["CrossingNumberGraph6B","Flexible"]

GraphData["CrossingNumberGraph6B","Graph","Flexible"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph6B","LCF"]

GraphData["CrossingNumberGraph6B","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph6B","LCFSignature"]

GraphData["CrossingNumberGraph6B","LCFNotations"]

GraphData["CrossingNumberGraph6B","Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Divisors[GraphData["CrossingNumberGraph6B","V"]]

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph6B"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph6B"],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

UnitDistance

GraphData["CrossingNumberGraph6B","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph6B"],Debug->True]//Timing

GraphData["CrossingNumberGraph6B","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance construction

LabelGraph@

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph6B","VertexList"],GraphData["CrossingNumberGraph6B","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["CrossingNumberGraph6B","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph6B"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph6B"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"CrossingNumberGraph6B",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph6B"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={,,,,};

Matchstick

GraphData["CrossingNumberGraph6B","Matchstick"]

GraphData["CrossingNumberGraph6B","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph6B","Edges"]]

3D All

GraphData["CrossingNumberGraph6B","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph6B","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph6B","Polyhedral"]

GraphData["CrossingNumberGraph6B","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph6C"

GraphData["CrossingNumberGraph6C","Graph","MinimalCrossing"]

GraphData["CrossingNumberGraph6C","Graph","LCF"]

GraphData["CrossingNumberGraph6C","Graph","UnitDistance"]

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

GraphData["CrossingNumberGraph6C","Graph","All"]

GraphPlot[GraphData["CrossingNumberGraph6C","EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph6C","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData["CrossingNumberGraph6C","Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph6C"],20],UnitDistanceGraphEmbeddingQ]

Print[{#,FindSubgraphUnitDistanceEmbedding[GraphData[#],"Results"->All]//Timing}]&@"CrossingNumberGraph6C"

UniqueEmbeddings[l]

gs={,,};

"CrossingNumberGraph7A"

Names

GraphData["CrossingNumberGraph7A","StandardName"]

GraphData["CrossingNumberGraph7A","StandardNames"]

Primary

GraphData["CrossingNumberGraph7A"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph7A","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph7A","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph7A","EdgeList"]]

Default embedding

GraphData["CrossingNumberGraph7A","V"]

GraphPlot[GraphData["CrossingNumberGraph7A","Edges"],GraphLayout->"HighDimensionalEmbedding"]

v=Join[CirclePoints[16],CirclePoints[{x,Pi/2},4],{{0,0}}];

Solve[v[[18,1]]==v[[10,1]],x]

v=Join[CirclePoints[16],CirclePoints[{Sin[((3 π)/16)],Pi/2},4]];
v=Join[v,v[[#]]/2&/@{19,20}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[{1,16},1],e[15,18,10],e[2,20,7],e[11,17,6],e[14,19,3],{1,12},{13,8},{16,5},{4,9},e[18,22,20],e[19,21,17],{21,22}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph7A","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["CrossingNumberGraph7A","Graph","Degenerate"]

Bilateral

GraphData["CrossingNumberGraph7A","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph7A","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph7A","Planar"]

GraphData["CrossingNumberGraph7A","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph7A",{"RCN","CN"}]

GraphData["CrossingNumberGraph7A","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph7A"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CrossingNumberGraph7A"],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["CrossingNumberGraph7A"],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph7A"],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph7A","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CrossingNumberGraph7A"],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union

VertexCount@

GraphPlot@

v={#/5,#2/7}&@@@{{0,0},{5,0},{1,1},{4,1},{1,2},{2,2},{3,2},{4,2},{1,3},{2,3},{3,3},{4,3},{1,4},{4,4},{1,5},{4,5},{1,6},{2,6},{3,6},{4,6},{0,7},{5,7}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[15,1,2,16],e[5,21,22,8],e[1,4,8,12,14,16,20,19,18,17,15,13,9,5,3,2],e[3,6,11,12],e[4,7,10,9],{6,7},{13,14},{11,18},{19,10},{17,21},{20,22}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph7A","Circulant"]

GraphData["CrossingNumberGraph7A","Graph","Circulant"]//StyleGraphs

Line graph

GraphData["CrossingNumberGraph7A","Line"]

LineGraphQ[GraphData["CrossingNumberGraph7A"]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "CrossingNumberGraph7A"&]

RootGraph[GraphData["CrossingNumberGraph7A"]]

Halin

GraphData["CrossingNumberGraph7A","Halin"]

GraphData["CrossingNumberGraph7A","Graph","Halin"]//StyleGraphs

Rigid

GraphData["CrossingNumberGraph7A","Rigid"]

GraphData["CrossingNumberGraph7A","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["CrossingNumberGraph7A","Flexible"]

GraphData["CrossingNumberGraph7A","Graph","Flexible"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph7A","LCF"]

GraphData["CrossingNumberGraph7A","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph7A","LCFSignature"]

GraphData["CrossingNumberGraph7A","LCFNotations"]

GraphData["CrossingNumberGraph7A","Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute all

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph7A"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph7A"],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData["CrossingNumberGraph7A","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph7A"],Debug->True]//Timing

GraphData["CrossingNumberGraph7A","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph7A","VertexList"],GraphData["CrossingNumberGraph7A","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["CrossingNumberGraph7A","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7A"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7A"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph7A","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

gs=UniqueEmbeddings[l]

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["CrossingNumberGraph7A","Matchstick"]

GraphData["CrossingNumberGraph7A","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph7A","Edges"]]

3D All

GraphData["CrossingNumberGraph7A","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph7A","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph7A","Polyhedral"]

GraphData["CrossingNumberGraph7A","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph7B"

GraphData["CrossingNumberGraph7B","StandardName"]

Primary

GraphData["CrossingNumberGraph7B"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph7B","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph7B","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph7B","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph7B","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph7B","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph7B","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph7B","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph7B","Planar"]

GraphData["CrossingNumberGraph7B","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph7B",{"RCN","CN"}]

GraphData["CrossingNumberGraph7B","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph7B"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph7B"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph7B"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph7B"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph7B","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph7B","Circulant"]

GraphData["CrossingNumberGraph7B","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph7B","Halin"]

GraphData["CrossingNumberGraph7B","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph7B","LCF"]

GraphData["CrossingNumberGraph7B","LCFSignature"]

GraphData["CrossingNumberGraph7B","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph7B"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph7B"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph7B","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph7B"],Debug->True]//Timing

GraphData["CrossingNumberGraph7B","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph7B","VertexList"],GraphData["CrossingNumberGraph7B","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph7B","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7B"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7B"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7B"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"CrossingNumberGraph7B",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph7B"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["CrossingNumberGraph7B","Matchstick"]

GraphData["CrossingNumberGraph7B","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph7B","Edges"]]

3D All

GraphData["CrossingNumberGraph7B","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph7B","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph7B","Polyhedral"]

GraphData["CrossingNumberGraph7B","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph7C"

GraphData["CrossingNumberGraph7C","StandardName"]

Primary

GraphData["CrossingNumberGraph7C"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph7C","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph7C","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph7C","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph7C","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph7C","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph7C","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph7C","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph7C","Planar"]

GraphData["CrossingNumberGraph7C","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph7C",{"RCN","CN"}]

GraphData["CrossingNumberGraph7C","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph7C"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph7C"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph7C"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph7C"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph7C","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph7C","Circulant"]

GraphData["CrossingNumberGraph7C","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph7C","Halin"]

GraphData["CrossingNumberGraph7C","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph7C","LCF"]

GraphData["CrossingNumberGraph7C","LCFSignature"]

GraphData["CrossingNumberGraph7C","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph7C"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph7C"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph7C","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph7C"],Debug->True]//Timing

GraphData["CrossingNumberGraph7C","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph7C","VertexList"],GraphData["CrossingNumberGraph7C","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph7C","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7C"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7C"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7C"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"CrossingNumberGraph7C",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph7C"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["CrossingNumberGraph7C","Matchstick"]

GraphData["CrossingNumberGraph7C","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph7C","Edges"]]

3D All

GraphData["CrossingNumberGraph7C","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph7C","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph7C","Polyhedral"]

GraphData["CrossingNumberGraph7C","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph7D"

GraphData["CrossingNumberGraph7D","StandardName"]

All

GraphData["CrossingNumberGraph7D","Graph","All"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph7D","LCFSignature"]

GraphData["CrossingNumberGraph7D","Graph","LCF"]//StyleGraphs

(lcf=LCF[GraphData["CrossingNumberGraph7D"],All])//Column

LCFGraph@@@lcf

UnitDistance

GraphData["CrossingNumberGraph7D","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph7D"],Debug->True]//Timing

GraphData["CrossingNumberGraph7D","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph7D","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph7D"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

{"CrossingNumberGraph7D",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph7D"],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData["CrossingNumberGraph7D","Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph7D",{"RCN","CN"}]

GraphData["CrossingNumberGraph7D","Graph","MinimalCrossing"]//StyleGraphs

3D

Graph3D[GraphData["CrossingNumberGraph7D","Edges"]]

Default embedding

g=Graph@GraphData["CrossingNumberGraph7D","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing Construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph7D"],"CrossingNumber"->7]]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

"CrossingNumberGraph8C"

GraphData["CrossingNumberGraph8C","StandardName"]

Primary

GraphData["CrossingNumberGraph8C"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph8C","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph8C","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph8C","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph8C","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph8C","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph8C","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph8C","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph8C","Planar"]

GraphData["CrossingNumberGraph8C","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph8C",{"RCN","CN"}]

GraphData["CrossingNumberGraph8C","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph8C"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph8C"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph8C"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph8C"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph8C","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph8C","Circulant"]

GraphData["CrossingNumberGraph8C","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph8C","Halin"]

GraphData["CrossingNumberGraph8C","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph8C","LCF"]

GraphData["CrossingNumberGraph8C","LCFSignature"]

GraphData["CrossingNumberGraph8C","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph8C"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph8C"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph8C","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph8C"],Debug->True]//Timing

GraphData["CrossingNumberGraph8C","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph8C","VertexList"],GraphData["CrossingNumberGraph8C","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph8C","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph8C"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph8C"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph8C"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding original

{"CrossingNumberGraph8C",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph8C"],TimeConstraint->300,"Results"->All]//Timing}

l={"CrossingNumberGraph8C",{17363.164713`,{{"DeGreyGraph1581",{194.767595`,{}}},{{"Hamming",{3,3}},{0.00748`,{}}},{{"TriangularHoneycombObtuseKnight",10},{58.414418`,{}}},{{"TriangularHoneycombObtuseKnight",11},{233.407764`,{}}}}}};

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

UnitDistance by FindSubgraphUnitDistanceEmbedding new

{"CrossingNumberGraph8C",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph8C"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["CrossingNumberGraph8C","Matchstick"]

GraphData["CrossingNumberGraph8C","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph8C","Edges"]]

3D All

GraphData["CrossingNumberGraph8C","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph8C","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph8C","Polyhedral"]

GraphData["CrossingNumberGraph8C","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph9A"

GraphData["CrossingNumberGraph9A","StandardName"]

Primary

GraphData["CrossingNumberGraph9A"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph9A","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph9A","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph9A","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph9A","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph9A","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph9A","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph9A","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph9A","Planar"]

GraphData["CrossingNumberGraph9A","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph9A",{"RCN","CN"}]

GraphData["CrossingNumberGraph9A","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph9A"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph9A"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph9A"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph9A"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph9A","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph9A","Circulant"]

GraphData["CrossingNumberGraph9A","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph9A","Halin"]

GraphData["CrossingNumberGraph9A","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph9A","LCF"]

GraphData["CrossingNumberGraph9A","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph9A","LCFSignature"]

GraphData["CrossingNumberGraph9A","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph9A"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph9A"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph9A","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph9A"],Debug->True]//Timing

GraphData["CrossingNumberGraph9A","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph9A","VertexList"],GraphData["CrossingNumberGraph9A","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph9A","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph9A"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph9A"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph9A"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"CrossingNumberGraph9A",l=FindSubgraphUnitDistanceEmbedding[GraphData["CrossingNumberGraph9A"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["CrossingNumberGraph9A","Matchstick"]

GraphData["CrossingNumberGraph9A","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph9A","Edges"]]

3D All

GraphData["CrossingNumberGraph9A","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph9A","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph9A","Polyhedral"]

GraphData["CrossingNumberGraph9A","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph10A"

GraphData["CrossingNumberGraph10A","StandardName"]

All

GraphData["CrossingNumberGraph10A","Graph","All"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph10A","LCFSignature"]

GraphData["CrossingNumberGraph10A","Graph","LCF"]//StyleGraphs

UnitDistance

GraphData["CrossingNumberGraph10A","Graph","UnitDistance"]//StyleGraphs

l=FindSubgraphIsomorphism[GraphData["CrossingNumberGraph10A"],"Results"->All]//Timing

gs=UniqueEmbeddings[l]

Matchstick

GraphData["CrossingNumberGraph10A","Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph10A",{"RCN","CN"}]

GraphData["CrossingNumberGraph10A","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing Construction

GraphPlot@MinimalCrossingGraph[GraphData["CrossingNumberGraph10A"],"CrossingNumber"->10]

"CrossingNumberGraph10B"

GraphData["CrossingNumberGraph10B","StandardName"]

Primary

GraphData["CrossingNumberGraph10B"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["CrossingNumberGraph10B","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["CrossingNumberGraph10B","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["CrossingNumberGraph10B","EdgeList"]]

Default embedding

GraphPlot@GraphData["CrossingNumberGraph10B","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["CrossingNumberGraph10B","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["CrossingNumberGraph10B","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["CrossingNumberGraph10B","Graph","Circular"]//StyleGraphs

Planar

GraphData["CrossingNumberGraph10B","Planar"]

GraphData["CrossingNumberGraph10B","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph10B",{"RCN","CN"}]

GraphData["CrossingNumberGraph10A","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CrossingNumberGraph10B"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["CrossingNumberGraph10B"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph10B"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CrossingNumberGraph10B"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["CrossingNumberGraph10B","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CrossingNumberGraph10B","Circulant"]

GraphData["CrossingNumberGraph10B","Graph","Circulant"]//StyleGraphs

Halin

GraphData["CrossingNumberGraph10B","Halin"]

GraphData["CrossingNumberGraph10B","Graph","Halin"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph10B","LCF"]

GraphData["CrossingNumberGraph10B","HamiltonianCycleCount"]

GraphData["CrossingNumberGraph10B","LCFSignature"]

GraphData["CrossingNumberGraph10B","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["CrossingNumberGraph10B"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["CrossingNumberGraph10B"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["CrossingNumberGraph10B","UnitDistance"]

UnitDistanceGraphQ[GraphData["CrossingNumberGraph10B"],Debug->True]//Timing

GraphData["CrossingNumberGraph10B","Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["CrossingNumberGraph10B","VertexList"],GraphData["CrossingNumberGraph10B","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph10B","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph10B"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph10B"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["CrossingNumberGraph10B"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

l=FindSubgraphIsomorphism[GraphData["CrossingNumberGraph10B"],"Results"->All]//Timing

gs=UniqueEmbeddings[l]

Matchstick

GraphData["CrossingNumberGraph10B","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["CrossingNumberGraph10B","Edges"]]

3D All

GraphData["CrossingNumberGraph10B","Graph",{"3D","All"}]

3D UnitDistance

GraphData["CrossingNumberGraph10B","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["CrossingNumberGraph10B","Polyhedral"]

GraphData["CrossingNumberGraph10B","Graph",{"3D","Polyhedron"}]

"CrossingNumberGraph12A"

GraphData["CrossingNumberGraph12A","StandardName"]

All

GraphData["CrossingNumberGraph12A","Graph","All"]//StyleGraphs

LCF

GraphData["CrossingNumberGraph12A","LCFSignature"]

GraphData["CrossingNumberGraph12A","Graph","LCF"]//StyleGraphs

UnitDistance

GraphData["CrossingNumberGraph12A","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["CrossingNumberGraph12A","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

l=FindSubgraphIsomorphism[GraphData["CrossingNumberGraph12A"],"Results"->All]//Timing

gs=UniqueEmbeddings[l]

Matchstick

GraphData["CrossingNumberGraph12A","Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData["CrossingNumberGraph12A",{"RCN","CN"}]

GraphData["CrossingNumberGraph12A","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing Construction

GraphPlot@MinimalCrossingGraph[GraphData["CrossingNumberGraph12A"],"CrossingNumber"->12]

"RefutedCrossingNumberGraph8D"

GraphData["RefutedCrossingNumberGraph8D","StandardName"]

Primary

GraphData["RefutedCrossingNumberGraph8D"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["RefutedCrossingNumberGraph8D","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["RefutedCrossingNumberGraph8D","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["RefutedCrossingNumberGraph8D","EdgeList"]]

Default embedding

GraphPlot@GraphData["RefutedCrossingNumberGraph8D","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["RefutedCrossingNumberGraph8D","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["RefutedCrossingNumberGraph8D","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["RefutedCrossingNumberGraph8D","Graph","Circular"]//StyleGraphs

Planar

GraphData["RefutedCrossingNumberGraph8D","Planar"]

GraphData["RefutedCrossingNumberGraph8D","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["RefutedCrossingNumberGraph8D",{"RCN","CN"}]

GraphData["RefutedCrossingNumberGraph8D","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["RefutedCrossingNumberGraph8D"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["RefutedCrossingNumberGraph8D"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["RefutedCrossingNumberGraph8D"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["RefutedCrossingNumberGraph8D"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["RefutedCrossingNumberGraph8D","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["RefutedCrossingNumberGraph8D","Circulant"]

GraphData["RefutedCrossingNumberGraph8D","Graph","Circulant"]//StyleGraphs

Halin

GraphData["RefutedCrossingNumberGraph8D","Halin"]

GraphData["RefutedCrossingNumberGraph8D","Graph","Halin"]//StyleGraphs

LCF

GraphData["RefutedCrossingNumberGraph8D","LCF"]

GraphData["RefutedCrossingNumberGraph8D","HamiltonianCycleCount"]

GraphData["RefutedCrossingNumberGraph8D","LCFSignature"]

GraphData["RefutedCrossingNumberGraph8D","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["RefutedCrossingNumberGraph8D"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["RefutedCrossingNumberGraph8D"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["RefutedCrossingNumberGraph8D","UnitDistance"]

UnitDistanceGraphQ[GraphData["RefutedCrossingNumberGraph8D"],Debug->True]//Timing

GraphData["RefutedCrossingNumberGraph8D","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["RefutedCrossingNumberGraph8D","VertexList"],GraphData["RefutedCrossingNumberGraph8D","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["RefutedCrossingNumberGraph8D","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8D"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8D"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8D"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"RefutedCrossingNumberGraph8D",l=FindSubgraphUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8D"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["RefutedCrossingNumberGraph8D","Matchstick"]

GraphData["RefutedCrossingNumberGraph8D","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["RefutedCrossingNumberGraph8D","Edges"]]

3D All

GraphData["RefutedCrossingNumberGraph8D","Graph",{"3D","All"}]

3D UnitDistance

GraphData["RefutedCrossingNumberGraph8D","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["RefutedCrossingNumberGraph8D","Polyhedral"]

GraphData["RefutedCrossingNumberGraph8D","Graph",{"3D","Polyhedron"}]

"RefutedCrossingNumberGraph8E"

GraphData["RefutedCrossingNumberGraph8E","StandardName"]

Primary

GraphData["RefutedCrossingNumberGraph8E"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["RefutedCrossingNumberGraph8E","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["RefutedCrossingNumberGraph8E","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["RefutedCrossingNumberGraph8E","EdgeList"]]

Default embedding

GraphPlot@GraphData["RefutedCrossingNumberGraph8E","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["RefutedCrossingNumberGraph8E","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["RefutedCrossingNumberGraph8E","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["RefutedCrossingNumberGraph8E","Graph","Circular"]//StyleGraphs

Planar

GraphData["RefutedCrossingNumberGraph8E","Planar"]

GraphData["RefutedCrossingNumberGraph8E","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["RefutedCrossingNumberGraph8E",{"RCN","CN"}]

GraphData["RefutedCrossingNumberGraph8E","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["RefutedCrossingNumberGraph8E"],TimeConstraint->10])//AbsoluteTiming

CrossingNumber[GraphData["RefutedCrossingNumberGraph8E"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["RefutedCrossingNumberGraph8E"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["RefutedCrossingNumberGraph8E"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["RefutedCrossingNumberGraph8E","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["RefutedCrossingNumberGraph8E","Circulant"]

GraphData["RefutedCrossingNumberGraph8E","Graph","Circulant"]//StyleGraphs

Halin

GraphData["RefutedCrossingNumberGraph8E","Halin"]

GraphData["RefutedCrossingNumberGraph8E","Graph","Halin"]//StyleGraphs

LCF

GraphData["RefutedCrossingNumberGraph8E","LCF"]

GraphData["RefutedCrossingNumberGraph8E","LCFSignature"]

GraphData["RefutedCrossingNumberGraph8E","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["RefutedCrossingNumberGraph8E"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["RefutedCrossingNumberGraph8E"],All])[[All,2]]]//Timing

gs=LCFGraph@@@lcf//StyleGraphs

gs=Join[With[{g=BilateralLCFEmbedding[#]},If[MissingQ[g],#,g]]&/@LCFGraph@@@DeleteCases[lcf,{_,1}],DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]]

UnitDistance

GraphData["RefutedCrossingNumberGraph8E","UnitDistance"]

UnitDistanceGraphQ[GraphData["RefutedCrossingNumberGraph8E"],Debug->True]//Timing

GraphData["RefutedCrossingNumberGraph8E","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["RefutedCrossingNumberGraph8E","VertexList"],GraphData["RefutedCrossingNumberGraph8E","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine/@Complement@@(GraphData["RefutedCrossingNumberGraph8E","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8E"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8E"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

FindUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8E"],MaxIterations->50000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"RefutedCrossingNumberGraph8E",l=FindSubgraphUnitDistanceEmbedding[GraphData["RefutedCrossingNumberGraph8E"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["RefutedCrossingNumberGraph8E","Matchstick"]

GraphData["RefutedCrossingNumberGraph8E","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["RefutedCrossingNumberGraph8E","Edges"]]

3D All

GraphData["RefutedCrossingNumberGraph8E","Graph",{"3D","All"}]

3D UnitDistance

GraphData["RefutedCrossingNumberGraph8E","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["RefutedCrossingNumberGraph8E","Polyhedral"]

GraphData["RefutedCrossingNumberGraph8E","Graph",{"3D","Polyhedron"}]