Frankl-Hubai-Palvolgyi Graph

Author

Eric W. Weisstein
July 26, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/Frankl-Hubai-PalvolgyiGraph.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Frankl-Hubai-PalvolgyiGraph.html.

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

Figure

GraphicsGrid[
Partition[
StyleGraphs[
GraphData["FranklHubaiPalvolgyiGraph","Graph","All"]
],
UpTo[3]
],
Dividers->All
]

Construction

From Ed Pegg, Jr.
May 2026

A 34 vertex graph without a 4-coloring if the origin is bichromatic.

https://arxiv.org/pdf/1912.02604 
Vertices 1 and 35 are degenerate at the origin. The distance 0 edge {1,35}  is to force chromatic number 5.
To get the 34 vertex version, map 35->1. I figured the degenerate version would be more useful.

vv=;
ee={{1,35},{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{1,11},{1,12},{1,13},{1,14},{1,15},{1,16},{1,17},{1,18},{1,19},{2,3},{2,7},{2,23},{2,35},{3,4},{3,22},{3,35},{4,5},{4,24},{4,35},{5,6},{5,20},{5,35},{6,7},{6,25},{6,35},{7,21},{7,35},{8,9},{8,13},{8,27},{8,35},{9,10},{9,20},{9,28},{9,34},{9,35},{10,11},{10,29},{10,35},{11,12},{11,21},{11,30},{11,32},{11,35},{12,13},{12,31},{12,35},{13,22},{13,26},{13,33},{13,35},{14,15},{14,19},{14,25},{14,31},{14,32},{14,35},{15,16},{15,26},{15,35},{16,17},{16,23},{16,27},{16,33},{16,35},{17,18},{17,28},{17,35},{18,19},{18,24},{18,29},{18,34},{18,35},{19,30},{19,35},{20,26},{20,33},{21,28},{21,34},{22,30},{22,32},{23,29},{23,34},{24,31},{24,32},{25,27},{25,33},{26,33},{27,33},{28,34},{29,34},{30,32},{31,32},{32,33},{32,34},{33,34}};
FranklHubaiPalvolgyiGraph=Graph[Range[35],ee,VertexCoordinates->Thread[Range[35]->vv],VertexLabels->Placed["Name",Center]]

RecognizeGraph@FranklHubaiPalvolgyiGraph

VertexChromaticNumber[g]

RootReduce[EuclideanDistance@@vv[[#]]]&/@ee

Embeddings

<<MathWorld`Graphs`

Names

GraphData["FranklHubaiPalvolgyiGraph","StandardName"]

GraphData["FranklHubaiPalvolgyiGraph","StandardNames"]

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

Wikidata

GraphData["FranklHubaiPalvolgyiGraph","WikidataID"]

GraphData["FranklHubaiPalvolgyiGraph","WikidataID","URL"]

Primary

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

All

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

CurvedEdge

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

Inexact

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

Degenerate

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

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

Embeddings

GraphDataEmbeddings[GraphData["FranklHubaiPalvolgyiGraph"]]

Default embedding

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

Configuration

GraphData["FranklHubaiPalvolgyiGraph","Configuration"]

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

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

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

Bipartite

GraphData["FranklHubaiPalvolgyiGraph","Bipartite"]

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

Bilateral

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

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

Circular

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

Circular construction 1*^4 [13 s: 0 bilateral]

Length[hc=FindHamiltonianCycle[GraphData["FranklHubaiPalvolgyiGraph"],1*^4][[All,All,1]]]//Timing

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

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

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

Circular construction 1*^5 [135 s: 0 bilateral]

Length[hc=FindHamiltonianCycle[GraphData["FranklHubaiPalvolgyiGraph"],1*^5][[All,All,1]]]//Timing

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

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

Circular construction 1*^6 [1316 s: 0 bilateral]

Length[hc=FindHamiltonianCycle[GraphData["FranklHubaiPalvolgyiGraph"],1*^6][[All,All,1]]]//Timing

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

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

Circular construction 1*^7 [12970 s: 987 bilateral]

Length[hc=FindHamiltonianCycle[GraphData["FranklHubaiPalvolgyiGraph"],1*^7][[All,All,1]]]//Timing

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

SymmetricHamiltonianCycleGraph[GraphData["FranklHubaiPalvolgyiGraph"],#]&/@Take[hcsym,50]

g=;

Circular construction All

GraphData["FranklHubaiPalvolgyiGraph","Hamiltonian"]

GraphData["FranklHubaiPalvolgyiGraph","HamiltonianCycles"]

GraphData["FranklHubaiPalvolgyiGraph","HamiltonianCycleCount"]

GraphData["FranklHubaiPalvolgyiGraph","LCF"]

GraphData["FranklHubaiPalvolgyiGraph","LCFSignature"]

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

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

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

Planar

GraphData["FranklHubaiPalvolgyiGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de

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

DateString[]

Quit

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["FranklHubaiPalvolgyiGraph","CN"]

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

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["FranklHubaiPalvolgyiGraph","RCN"]

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

MinimalCrossing construction Tutte

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

(g=MinimalCrossingGraph[GraphData["FranklHubaiPalvolgyiGraph"],"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["FranklHubaiPalvolgyiGraph","V"]])]
}&/@TutteGraphs;

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

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

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

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

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

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

MinimalCrossing construction Random 1k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["FranklHubaiPalvolgyiGraph"],
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

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 [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["FranklHubaiPalvolgyiGraph"],
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["FranklHubaiPalvolgyiGraph"],TimeConstraint->1])//AbsoluteTiming

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["FranklHubaiPalvolgyiGraph"],
20*^3,
"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 1k [old]

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

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

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["FranklHubaiPalvolgyiGraph","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["FranklHubaiPalvolgyiGraph","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["FranklHubaiPalvolgyiGraph"],TimeConstraint->1])//AbsoluteTiming

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

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["FranklHubaiPalvolgyiGraph","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["FranklHubaiPalvolgyiGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

GraphData["FranklHubaiPalvolgyiGraph","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["FranklHubaiPalvolgyiGraph"],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData["FranklHubaiPalvolgyiGraph"],"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["FranklHubaiPalvolgyiGraph"],TimeConstraint->10])//AbsoluteTiming

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

Circulant

GraphData["FranklHubaiPalvolgyiGraph","Circulant"]

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

Circulant construction

GraphData["FranklHubaiPalvolgyiGraph","NotationRules","Circulant"]

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

CirculantGraph1@@@ci

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

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["FranklHubaiPalvolgyiGraph","Line"]

LineGraphQ[GraphData["FranklHubaiPalvolgyiGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["FranklHubaiPalvolgyiGraph"]]

Halin

GraphData["FranklHubaiPalvolgyiGraph","Halin"]

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

Rigid

GraphData["FranklHubaiPalvolgyiGraph","Rigid"]

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

Flexible

GraphData["FranklHubaiPalvolgyiGraph","Flexible"]

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

Voltage

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

Voltage construction

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

Length[gs]

RecognizeGraph/@gs

VoltageGraphNotation/@gs//Column

EmbeddingClasses/@gs//Column

GraphSymmetryGroup/@gs//Column

LCF

GraphData["FranklHubaiPalvolgyiGraph","LCF"]

GraphData["FranklHubaiPalvolgyiGraph","HamiltonianCycleCount"]

GraphData["FranklHubaiPalvolgyiGraph","LCFSignature"]

GraphData["FranklHubaiPalvolgyiGraph","LCFNotations"]

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

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

UnitDistance

GraphData["FranklHubaiPalvolgyiGraph","UnitDistance"]

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

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

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

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

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[Graph[VertexList[#],EdgeList[#],VertexCoordinates->GraphEmbedding[#]],MaxIterations->500]&/@Complement@@(GraphData["FranklHubaiPalvolgyiGraph","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["FranklHubaiPalvolgyiGraph"],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["FranklHubaiPalvolgyiGraph"],"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData["FranklHubaiPalvolgyiGraph","Matchstick"]

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

3D default

Graph3D[GraphData["FranklHubaiPalvolgyiGraph","Edges"]]

3D construction

3D All

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

3D UnitDistance construction Random

GraphData["FranklHubaiPalvolgyiGraph","Dimension"]

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

EdgeLengths[g]

UnitDistanceGraphEmbeddingQ[g]

3D UnitDistance

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

3D Polyhedron

GraphData["FranklHubaiPalvolgyiGraph","Polyhedral"]

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

GraphData

GraphDataClassString

GraphDataClassString["FranklHubaiPalvolgyiGraph"]

GraphDataString

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

WriteString[
outstream,
GraphDataString[
FranklHubaiPalvolgyiGraph,
"StandardName"->"FranklHubaiPalvolgyiGraph",
"Name"->"Frankl\[Hyphen]Hubai\[Hyphen]Palvolgyi graph"
]
];

Close[outstream]