Cameron-Montanaro-Newman-Severini-Winter Graph

Author

Eric W. Weisstein
June 9, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/Cameron-Montanaro-Newman-Severini-WinterGraph.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Cameron-Montanaro-Newman-Severini-WinterGraph.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Source

Cameron, P. J.; Montanaro, A.; Newman, M. W.; Severini, S.; and Winter, A. "On the Quantum Chromatic Number of a Graph." Electronic Journal of Combinatorics 14, No. 1, R81, 2007. https://doi.org/10.37236/999.

Figure

Annotate[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],ImageSize->200]//StyleGraphs

Embeddings

<<MathWorld`Graphs`

Names

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","StandardName"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","StandardNames"]

HouseOfGraphs

hogid=GraphData["CameronMontanaroNewmanSeveriniWinterGraph","NotationRules","HouseOfGraphs"]

HouseOfGraphsLookup[hogid]

HouseOfGraphsGraph[hogid]//StyleGraphs[#,ImageSize->Small]&

Wikidata

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","WikidataID"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","WikidataID","URL"]

Primary

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

All

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

CurvedEdge

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","CurvedEdge"]//StyleGraphs

Inexact

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

Degenerate

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

DegenerateGraphEmbeddingTypes/@GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","Degenerate"]//Column

Embeddings

GraphDataEmbeddings[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"]]

Default embedding

g0=GraphPlot[Graph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","VertexList"],GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Edges"]]]

Configuration

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Configuration"]

configurations=GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","Configuration"]//StyleGraphs

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","CompleteKPartite"]//StyleGraphs

Bipartite

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Bipartite"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","Bipartite"]//StyleGraphs

Bilateral

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

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]//Quiet

Circular

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

Circular construction All

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Hamiltonian"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","HamiltonianCycles"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","HamiltonianCycleCount"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","LCF"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","LCFSignature"]

Length[hc=FindHamiltonianCycle[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],All][[All,All,1]]]//Timing

g=CircularEmbeddingFromHamiltonianCycle[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],hc[[1]]]

RecognizeGraph@g

Length[hcsym=SymmetricHamiltonianCycles[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],hc,ProgressReporting->True]]//Timing

GroupBy[SymmetricHamiltonianCycleGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],#]&/@hcsym,GraphSymmetryGroup]

Planar

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de [verified 16]

CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],Method->"WebCompute"]//AbsoluteTiming

DateString[]

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->1,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->1,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->1,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->1,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->1,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber: "EmbeddingScheme" -> 2

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->2,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->2,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->2,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->2,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->2,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->2,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber: "EmbeddingScheme" -> 3

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->3,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->3,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->3,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->3,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->3,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"EmbeddingScheme"->3,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossing

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","CN"]

(gs=GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","MinimalCrossing"])//StyleGraphs

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","RCN"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","MinimalRectilinearCrossing"]//StyleGraphs

MinimalCrossing construction Tutte

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

TutteGraphs=BilateralGraph[CenterGraph@IGLayoutTutte[g,"OuterFace"->#],"ReturnNonbilateralGraph"->True]&/@faces

TutteGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]])]
}&/@TutteGraphs;

Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

LocalCrossingNumberWithEmbeddings=
{1+Max[Length/@DeleteCases[
Drop[AdjacencyList[#],GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]],
_Integer?(LessEqualThan[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]]),
{2}
]],#}&/@TutteGraphs

TutteGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@TutteGraphs;

SymmetricEmbeddingsWithOrientations=DeleteCases[
TutteGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]

Select[SymmetricEmbeddingsWithOrientations[[All,1]],Count[OuterFace[#],_?(GreaterThan[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]])]==0&]

MinimalCrossing construction Random 1k [213 s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],
1000,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

diagrams1

g=;

GraphCrossingCount@g

GraphCrossingDiagram@g

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 5k [1100 s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],
5000,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 20k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],
20*^3,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

g=;

GraphCrossingCount@g

GraphCrossingDiagram@g

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 1k [old]

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

(RandomGraphs=Table[MinimalCrossingGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"CrossingNumber"->cn,RandomSeed->n],{n,1000}]//Union);//Timing

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]])]
}&/@RandomGraphs;

Small numbers of crossing points on the boundary:

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]/.g_Graph:>BilateralGraph[CenterGraph@g,"ReturnNonbilateralGraph"->True]

MinimalCrossing construction Random 20k [old]

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

(RandomGraphs=Table[MinimalCrossingGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"CrossingNumber"->cn,RandomSeed->n],{n,20000}]//Union);//Timing

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]])]
}&/@RandomGraphs;

Small numbers of crossing points on the boundary:

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["CameronMontanaroNewmanSeveriniWinterGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]/.g_Graph:>BilateralGraph[CenterGraph@g,"ReturnNonbilateralGraph"->True]

MinimalLocalCrossing

GraphData["CameronMontanaroNewmanSeveriniWinterGraph",{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalRectilinearLocalCrossing

GraphData["CameronMontanaroNewmanSeveriniWinterGraph",{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","MinimalRectilinearLocalCrossing"]//StyleGraphs

MinimaRectilinearlLocalCrossing construction 1000

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

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

(gs=Select[Table[MinimalCrossingGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,1000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

MinimaRectilinearlLocalCrossing construction 20000

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

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

(gs=Select[Table[MinimalCrossingGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Circulant

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Circulant"]

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

Circulant construction

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","NotationRules","Circulant"]

(ci=CirculantIndices[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"]])//AbsoluteTiming

CirculantGraph1@@@ci

Cases[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Line"]

LineGraphQ[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"]]

Halin

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Halin"]

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

Rigid

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Rigid"]

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

Flexible

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Flexible"]

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

Voltage

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph","Voltage"]//StyleGraphs

Voltage construction

gs=VoltageGraph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"MaxResults"->20,Debug->True]

Length[gs]

RecognizeGraph/@gs

VoltageGraphNotation/@gs//Column

EmbeddingClasses/@gs//Column

GraphSymmetryGroup/@gs//Column

LCF

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","LCF"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","HamiltonianCycleCount"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","LCFSignature"]

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","LCFNotations"]

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

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

UnitDistance

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","UnitDistance"]

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","VertexList"],GraphData["CameronMontanaroNewmanSeveriniWinterGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[Graph[VertexList[#],EdgeList[#],VertexCoordinates->GraphEmbedding[#]],MaxIterations->500]&/@Complement@@(GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Graph",#]&/@{"All","UnitDistance"})

TextGrid[
gdata=SortBy[
{#,DegenerateGraphEmbeddingQ[#],GraphSymmetryGroup[#],Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
],
Dividers->All
]

Cases[gdata,{__,0.}]

Symmetric:

gs=Cases[
gdata,
{g_,_,sym:Except[TransformationGroup[{}]],diff_}:>{
HighlightGraph[g1=BilateralGraph[CenterGraph[g],"ReturnNonbilateralGraph"->True],Flatten[FindCoincidentVertices[g1]]],
sym,
diff
}]

UnitDistance by FindUnitDistanceEmbedding  (random embeddings returning on messages)

gs=FindUnitDistanceEmbedding[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],50,"ReturnWithMessage"->True];//Quiet//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

SymmetricEmebddings=Select[gdata,GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]&]

UnitDistance by FindUnitDistanceEmbedding  (random embedding waiting until UD is found)

FindUnitDistanceEmbedding[GraphData["CameronMontanaroNewmanSeveriniWinterGraph"],"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Matchstick"]

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

3D default

Graph3D[GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Edges"]]

3D construction

3D All

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

3D UnitDistance construction Random

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Dimension"]

(g=UnitDistanceEmbeddingRefine[
Graph3D[
GraphData["CameronMontanaroNewmanSeveriniWinterGraph","VertexList"],
GraphData["CameronMontanaroNewmanSeveriniWinterGraph","EdgeList"]
],
WorkingPrecision->25])//Timing

EdgeLengths[g]

UnitDistanceGraphEmbeddingQ[g]

3D UnitDistance

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

3D Polyhedron

GraphData["CameronMontanaroNewmanSeveriniWinterGraph","Polyhedral"]

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

Construction

CameronMontanaroNewmanSeveriniWinterVectors is the list of 18 four-dimensional vectors associated with the 18 vertices of the Cameron-Montanaro-Newman-Severini-Winter graph in Cameron-Montanaro-Newman-Severini-WinterGraph.nb (line 179).

Vertex i gets vector vectors[[i]]; the graph edges are between orthogonal vector pairs. For example vertices 1 and 2 are adjacent, and {0,0,1,-1}.{1,0,0,0} == 0.

CameronMontanaroNewmanSeveriniWinterVectors = {
	{0, 0, 1, -1}, {1, 0, 0, 0}, {0, 1, 1, 1}, {0, 1, 0, -1},
	{0, 0, 1, 0}, {1, 1, 0, 1}, {1, -1, 0, 0}, {0, 0, 0, 1},
	{1, 1, 1, 0}, {1, 0, -1, 0}, {0, 1, 0, 0}, {1, 0, 1, 1},
	{0, 1, -1, 0}, {1, 0, 0, -1}, {1, 1, 1, 1}, {1, 1, -1, -1},
	{1, -1, 1, -1}, {1, -1, -1, 1}
};

CameronMontanaroNewmanSeveriniWinterGraph = Graph[
	Range[18],
	UndirectedEdge @@@ {
		{1, 2}, {1, 3}, {1, 11}, {1, 12}, {1, 16},
		{2, 3}, {2, 4}, {2, 13}, {3, 4}, {3, 13},
		{4, 5}, {4, 6}, {4, 10}, {4, 17}, {5, 6},
		{5, 7}, {5, 14}, {6, 7}, {6, 14}, {7, 8},
		{7, 9}, {7, 16}, {8, 9}, {8, 10}, {8, 13},
		{9, 10}, {9, 13}, {10, 11}, {10, 12}, {10, 17},
		{11, 12}, {11, 14}, {12, 14}, {13, 14}, {13, 15},
		{13, 18}, {14, 15}, {14, 18}, {15, 16}, {15, 17},
		{15, 18}, {16, 17}, {16, 18}, {17, 18}
	},
	VertexLabels -> Automatic
]

QuantumChromaticNumber[CameronMontanaroNewmanSeveriniWinterGraph,Debug->True]//AbsoluteTiming

GraphData

GraphDataClassString

GraphDataClassString["CameronMontanaroNewmanSeveriniWinterGraph"]

GraphDataString

outstream=OpenWrite["~/CameronMontanaroNewmanSeveriniWinterGraph.m"]

WriteString[
outstream,
GraphDataString[
CameronMontanaroNewmanSeveriniWinterGraph,
"StandardName"->"CameronMontanaroNewmanSeveriniWinterGraph",
"Name"->"Cameron\[Hyphen]Montanaro\[Hyphen]Newman\[Hyphen]Severini\[Hyphen]Winter graph",
"Information"->"Cameron-Montanaro-Newman-Severini-Winter Graph",
"QuantumChromaticNumber"->4
]
];

Close[outstream]