Barnette-Bosák-Lederberg Graph

Author

Eric W. Weisstein
January 18, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/Barnette-Bosak-LederbergGraph.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Barnette-Bosak-LederbergGraph.html.

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

Figures

Read and Wilson + Grünbaum

GraphicsRow[Take[GraphData["BarnetteBosakLederbergGraph","Graph","All"],2]//StyleGraphs,Dividers->All]

Other

GraphicsRow[Take[GraphData["BarnetteBosakLederbergGraph","Graph","All"],{3,6}]//StyleGraphs,Dividers->All]

Quit

Embeddings

<<MathWorld`Graphs`

Names

GraphData["BarnetteBosakLederbergGraph","StandardName"]

GraphData["BarnetteBosakLederbergGraph","StandardNames"]

Primary

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

Read and Wilson construction

g=Graph[Range[38],UndirectedEdge@@@{{1,2},{1,3},{1,11},{2,5},{2,20},{3,4},{3,6},{4,5},{4,8},{5,10},{6,7},{6,12},{7,8},{7,13},{8,9},{9,10},{9,14},{10,15},{11,12},{11,16},{12,13},{13,14},{14,15},{15,18},{16,17},{16,21},{17,18},{17,22},{18,19},{19,20},{19,23},{20,38},{21,22},{21,24},{22,23},{23,28},{24,25},{24,37},{25,26},{25,29},{26,27},{26,30},{27,28},{27,32},{28,33},{29,30},{29,34},{30,31},{31,32},{31,35},{32,33},{33,36},{34,35},{34,37},{35,36},{36,38},{37,38}},VertexCoordinates->(10^(-4)){{6977,-19679},{6977,-2656},{8819,-18615},{8819,-11167},{8819,-3720},{10661,-17551},{10661,-13295},{10661,-11167},{10661,-9039},{10661,-4784},{12503,-19679},{12503,-16487},{12503,-13295},{12503,-9039},{12503,-5848},{14346,-19679},{14346,-11167},{14346,-4784},{16188,-3720},{16188,-2656},{18030,-19679},{18030,-11167},{18030,-4784},{19872,-19679},{19872,-16487},{19872,-13295},{19872,-9039},{19872,-5848},{21714,-17551},{21714,-13295},{21714,-11167},{21714,-9039},{21714,-4784},{23556,-18615},{23556,-11167},{23556,-3796},{25398,-19679},{25398,-2656}},VertexLabels->Automatic]

RecognizeGraph@g

Grünbaum construction

GraphPlot[RotateGraph[,Pi/2],VertexSize->.5,PerformanceGoal->"Speed"]

g0=LabelGraph@CenterGraph@MyGraphUnion[
Annotate[g1=RotateGraph[GraphData["TutteFragment"], -Pi/2],VertexCoordinates->({#,-#2}&@@@GraphEmbedding[g1])],
Annotate[g2=RotateGraph[GraphData["TutteFragment"], Pi/2],VertexCoordinates->({#,#2}&@@@GraphEmbedding[g2])],
Offset->2/3
]

v=GraphEmbedding[g0];
v=Join[v,Subdivide[v[[2]],v[[17]],4][[2;;-2]],{Mean[v[[{3,18}]]]}];
v=Join[v,{v[[#,1]],v[[1,2]]}&/@{31,33}];
v=Join[v,Mean[v[[#]]]&/@{{31,35},{33,36}}];
ee=Join[EdgeList[g0],{e[3,34,18],e[1,35,36,16],e[35,37,31],e[36,38,33],{32,34},e[2,31,32,33,17],{37,38}}];
g=Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

All

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

Inexact

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

Embeddings

GraphDataEmbeddings[GraphData["BarnetteBosakLederbergGraph"]]

Default embedding

GraphPlot[Graph[GraphData["BarnetteBosakLederbergGraph","VertexList"],GraphData["BarnetteBosakLederbergGraph","Edges"]]]

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

g=FindCorrespondingGraph[,v]

RecognizeGraph@g

DegenerateGraphEmbeddingTypes@g

Degenerate

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

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

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

GeneralizedPetersen

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

IGraph

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

Bilateral

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

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

Circular

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

Circular construction

GraphData["BarnetteBosakLederbergGraph","Hamiltonian"]

GraphData["BarnetteBosakLederbergGraph","LCF"]

GraphData["BarnetteBosakLederbergGraph","HamiltonianCycleCount"]

Planar

GraphData["BarnetteBosakLederbergGraph","Planar"]

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

Planar construction

PlanarGraphEmbedding[GraphData["BarnetteBosakLederbergGraph"],GraphLayout->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData["BarnetteBosakLederbergGraph"]],10]

PlanarGraphEmbeddings

GraphData["BarnetteBosakLederbergGraph","PlanarEmbeddingCount"]

(plemb=PlanarGraphEmbeddings[GraphData["BarnetteBosakLederbergGraph"],Debug->True])//Timing

Length[plemb]

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData["BarnetteBosakLederbergGraph"]]

Tally[ReverseSort[Length/@faces]]

gs=IGLayoutTutte[GraphData["BarnetteBosakLederbergGraph"],"OuterFace"->#]&/@faces

TextGrid[DeleteCases[{#,GraphSymmetryGroup[#]}&/@gs,{_,TransformationGroup[{}]}]/.g_Graph:>BilateralGraph[g,"ReturnNonbilateralGraph"->True],Dividers->All]

Grid[DeleteCases[{#,Cases[GraphSymmetryGroup[#],r_Rotation:>r,∞]}&/@gs,{_,{}}],Dividers->All]

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

gs=UnlabelGraph/@{,,,}

Tutte embedding construction 9

GraphPlot[,VertexSize->.7,PerformanceGoal->"Speed"]

v={#/14,#2/6}&@@@Flatten[MapIndexed[Thread[{#,#2[[1]]-1}]&,{{0,6,8,14},Delete[Range[13],7],{1,2,12,13},{0,1,13,14},{1,2,3,11,12,13},{1,4,7,10,13},{0,7,14}}],1];
ee={e[1,2,3,4,24,38,37,36,21,1,{5,16},4],{2,10},{3,11},e[9,33,12],e[5,17,22,25,31,36],e[16,20,23,30,35,38],e[31,32,8],e[13,34,35],e[25,26,27,7],e[14,28,29,30],{27,32},{28,34},{21,22},{23,24},e[6,18,26],e[15,19,29],{17,18},{19,20},{33,37}};
g=Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

GraphPlot@UnlabelGraph@

v={#/12,#2/6}&@@@Flatten[MapIndexed[Thread[{#,#2[[1]]-1}]&,{{0,5,7,12},Delete[Range[11],6],{1,2,10,11},{0,1,11,12},{1,2,3,9,10,11},{1,4,5,6,7,8,11},{0,6,12}}],1];
ee={e[1,2,3,4,22,38,37,36,19,1,5,15,20,23,29,36],e[4,14,18,21,28,35,38],e[2,9,31,32,33,10,3],e[29,30,8],e[11,34,35],{9,10},{32,37},e[23,24,25,30],e[34,26,27,28],e[5,6,7,8,31],e[33,11,12,13,14],{15,16},{17,18},e[6,16,24],e[13,17,27],{12,26},{7,25},{19,20},{21,22}};
g=Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

Tutte embedding construction 5

GraphPlot[,VertexSize->.8,PerformanceGoal->"Speed"]

v={#/14,#2/6}&@@@Flatten[MapIndexed[Thread[{#,#2[[1]]-1}]&,{{0,14},{6,8},{1,5,6,8,9,13},{3,5,9,11},{1,2,3,4,5,6,8,9,10,11,12,13},{1,2,4,6,7,8,10,12,13},{0,7,14}}],1];
ee={e[1,2,38,37,36,1,3,4,2],e[36,{27,35},38],e[27,15,5,6,7],e[35,26,10,9,8],e[3,7,20,30],e[4,8,21,32],{31,37},e[{15,20}],e[{21,26}],e[17,11,12],e[24,14,13],e[6,12,19],e[9,13,22],{18,29},{23,33},{25,34},{16,28},{5,11},{10,14}};
g=Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Tutte embedding construction 4

GraphPlot[,VertexSize->2,PerformanceGoal->"Speed"]

v={#/10,#2/8}&@@@Flatten[MapIndexed[Thread[{#,#2[[1]]-1}]&,{{0,10},{1,4,6,9},{1,2,3,7,8,9},{1,2,8,9},{2,3,7,8},{1,2,8,9},{1,2,3,4,5,6,7,8,9},{1,5,9},{0,10}}],1];
ee={e[1,2,38,37,1,3,7,13,21,25,34,37],e[2,6,12,16,24,33,36,38],e[34,35,36],e[3,4,28],e[30,5,6],e[{25,33}],{29,35},e[7,8,9,18,27],e[31,19,10,11,12],{4,9},{5,10},{17,18},{19,20},e[8,14,17,22,26],e[11,15,20,23,32],{21,22},{23,24},{15,16},{13,14}};
g=Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing

GraphData["BarnetteBosakLederbergGraph","CN"]

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

MinimalRectilinearCrossing

GraphData["BarnetteBosakLederbergGraph","RCN"]

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

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

Circulant

GraphData["BarnetteBosakLederbergGraph","Circulant"]

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

Circulant construction

GraphData["BarnetteBosakLederbergGraph","NotationRules","Circulant"]

(ci=CirculantIndices[GraphData["BarnetteBosakLederbergGraph"]])//Timing

CirculantGraph1@@@ci

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

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["BarnetteBosakLederbergGraph","Line"]

LineGraphQ[GraphData["BarnetteBosakLederbergGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["BarnetteBosakLederbergGraph"]]

Halin

GraphData["BarnetteBosakLederbergGraph","Halin"]

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

Rigid

GraphData["BarnetteBosakLederbergGraph","Rigid"]

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

Flexible

GraphData["BarnetteBosakLederbergGraph","Flexible"]

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

LCF

GraphData["BarnetteBosakLederbergGraph","LCF"]

GraphData["BarnetteBosakLederbergGraph","HamiltonianCycleCount"]

GraphData["BarnetteBosakLederbergGraph","LCFSignature"]

GraphData["BarnetteBosakLederbergGraph","LCFNotations"]

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

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

UnitDistance

GraphData["BarnetteBosakLederbergGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Select[IndexGraph/@gs,!DegenerateGraphEmbeddingQ[#]&&GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]//Timing

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

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

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

gs=UniqueEmbeddings[l]

gs={,,,,};

Matchstick

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

3D default

Graph3D[GraphData["BarnetteBosakLederbergGraph","Edges"]]

3D All

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

3D Polyhedron

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

Graphics3D[{Opacity[.5],Red,GraphData["BarnetteBosakLederbergGraph","Polyhedron"],Yellow,Sphere[]}]

3D Polyhedron construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData["BarnetteBosakLederbergGraph"],1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->2000,PrecisionGoal->20,"ProgressInterval"->500,"TangentFactor"->1/20,"PlanarFactor"->1/20,Debug->True])//Timing

Graphics3D[can]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/100,"PlanarFactor"->1/100,Debug->True])//Timing

Graphics3D[can]

(can=FindCanonicalPolyhedron[can,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/100,"PlanarFactor"->1/100,Debug->True])//Timing

Graphics3D[can=Polyhedron[]]

(can=FindCanonicalPolyhedron[can,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/500,"PlanarFactor"->1/500,Debug->True])//Timing

(can=FindCanonicalPolyhedron[can,WorkingPrecision->50,MaxIterations->100000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/500,"PlanarFactor"->1/500,Debug->True])//Timing

(can=FindCanonicalPolyhedron[can,WorkingPrecision->50,MaxIterations->200000,PrecisionGoal->20,"ProgressInterval"->5000,"TangentFactor"->1/500,"PlanarFactor"->1/500,Debug->True])//Timing

can=Polyhedron[];

Graphics3D[can]

SimplePolyhedronQ[can]

If[%===False,Quit[]]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors={Red,Green,Black,Blue};

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

3D Canonical construction from PolyhedralEmbedding

GraphData["BarnetteBosakLederbergGraph","Graph","Planar"]

g=GraphData["BarnetteBosakLederbergGraph","Graph","Planar"][[1]]

g=PlanarGraph[GraphData["BarnetteBosakLederbergGraph","Edges"]]

Show[g2=PolyhedralEmbedding[g,"Scale"->2],Boxed->True]//Quiet

With[
{faces=PlanarFaceList[Graph[EdgeList[g2]]],v=GraphEmbedding@g2},
Graphics3D[{Point[{0,0,0}],Opacity[.5],Sphere[],FaceForm[Green,Red],
poly=Polyhedron[1.02v,faces]}]]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->100,MaxIterations->1000,PrecisionGoal->20,"ProgressInterval"->100,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

Graphics3D[can]

SimplePolyhedronQ[can]

If[%===False,Quit[]]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

Embedding construction

Grünbaum

Grünbaum, Long paths in polytopes, Fig. 17.1.5

GraphPlot[Graph[List/@Sort[Sort/@DeleteCases[List@@@{1->1,2->2,3->3,4->4,5->5,6->6,7->7,8->8,9->9,10->10,11->11,12->12,13->13,14->14,13->5,5->6,14->7,7->4,4->2,2->1,12->11,11->10,10->3,3->2,12->13,13->14,6->7,5->8,8->9,8->11,9->10,15->15,16->16,16->15,15->3,15->4,9->16,16->6},{n_,n_}]],List/@{{570,510},{568,549},{525,581},{619,586},{561,674},{610,653},{684,655},{521,662},{533,642},{473,607},{433,660},{383,735},{566,750},{751,749},{565,598},{566,626}}/700.],Method->None]

g=ToCombinatoricaGraph[]

v=(#-{0,.65})&/@Vertices[g]

e=Edges[g]

(g2=AddEdges[AddVertices[AddEdges[AddVertices[ChangeVertices[g,v],{1,-1}#&/@v],e+16],{{.81,0},{1.07,0},{.56,.08},{.56,-.08},{.69,.08},{.69,-.08}}],{{14,34},{34,30},{1,33},{33,17},{33,34},{35,12},{35,36},{36,28},{35,37},{37,1},{37,38},{36,38},{38,17}}])//ShowLabeledGraph

RecognizeGraph[g2]

ShowGraph[g2,ImageSize->150]

Append[GraphData["BarnetteBosakLederbergGraph","Embeddings"],ToCommonEdges[g2,Graph["BarnetteBosakLederbergGraph"]]]//PadVertices

GraphicsArray[{ShowGraph[#]&/@{Graph["BarnetteBosakLederbergGraph"],g2}/.{PointSize[x_]:>AbsolutePointSize[5],
Thickness[x_]:>AbsoluteThickness[1]}}]

Holton and McKay

https://users.cecs.anu.edu.au/~bdm/papers/c4cp.pdf

v={{-1,0},{1,0},{0,Sqrt[3]},{-1/3,0},{0,0},{2/3,0},{-1/3,1/4},{0,1/4},{1/3,1/4},{2/3,1/4},{-1/3,1/2},{1/3,1/2}};
v=Join[v,Subdivide[v[[2]],v[[3]],3][[2;;-2]],Subdivide[v[[1]],v[[3]],3][[{3}]]];
ee={e[1,4,5,6,2,13,14,3,15,1],e[4,7,11,15],e[7,8,9,10,13],{6,10},{5,8},e[11,12,9],{12,14}};
TutteFragment=VertexReplace[Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic],{3->1,1->3}]

RecognizeGraph@TutteFragment

SpliceFragment[g_Graph,frag_Graph,{v0_Integer,l_}]:=Module[
{
v,ee,
n=Max[VertexList[g]],
f=VertexCount[frag],
gdel=VertexDelete[g,v0]
},
v=Join[
VertexList[gdel],
Range[n+1,n+f]
];
ee=Join[
EdgeList[gdel],
EdgeList[frag]/.i_Integer:>i+n,
UndirectedEdge@@@Transpose[{l,Range[Length[l]]+n}]
];
Graph[v,UndirectedEdge@@@ee]
]

SpliceFragment[g_Graph,frag_Graph,l_List]:=Fold[SpliceFragment[#,frag,#2]&,g,l]

prism=GraphData[{"Prism",5},"Graph","Labeled"]

CubicNonhamiltonian38=SpliceFragment[prism,TutteFragment,#]&/@{
{{9,{10,4,8}},{7,{6,2,8}}},
{{9,{8,4,10}},{7,{8,2,6}}},
{{9,{8,4,10}},{7,{6,2,8}}},
{{9,{10,4,8}},{2,{1,7,3}}},
{{9,{8,4,10}},{2,{3,7,1}}},
{{9,{8,4,10}},{2,{1,7,3}}}
}

VertexCount/@CubicNonhamiltonian38

RecognizeGraph/@CubicNonhamiltonian38

Select[Subsets[Range[6],{2}],IsomorphicGraphQ@@CubicNonhamiltonian38[[#]]&]

CubicGraphQ/@CubicNonhamiltonian38

HamiltonianGraphQ/@CubicNonhamiltonian38

PlatypusGraphQ/@CubicNonhamiltonian38

GraphData

GraphDataClassString

GraphDataClassString["BarnetteBosakLederbergGraph"]

GraphDataMatrices

GraphDataMatrices["BarnetteBosakLederbergGraph"]

GraphDataTable

TableForm[GraphDataTable["BarnetteBosakLederbergGraph",x],TableDepth->2]//TraditionalForm