Descartes Snarks

Author

Eric W. Weisstein
November 18, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Figure

Show[GraphData["DescartesSnark1"]//StyleGraphs[#,VertexSize->.7]&,ImageSize->500]

Embeddings

<<MathWorld`Graphs`

"DescartesSnark1"

Names

GraphData["DescartesSnark1","StandardName"]

GraphData["DescartesSnark1","StandardNames"]

Primary

GraphData["DescartesSnark1"]//StyleGraphs[#,ImageSize->Small,VertexSize->1]&

All

GraphData["DescartesSnark1","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["DescartesSnark1","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["DescartesSnark1","EdgeList"]]

Default embedding

GraphPlot@GraphData["DescartesSnark1","Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData["DescartesSnark1","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["DescartesSnark1","Graph","Degenerate"]

Perspective

GraphData["DescartesSnark1","Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData["DescartesSnark1","Polyhedron"]

Bilateral

GraphData["DescartesSnark1","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["DescartesSnark1","Graph","Circular"]//StyleGraphs

Planar

GraphData["DescartesSnark1","Planar"]

GraphData["DescartesSnark1","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["DescartesSnark1",{"RCN","CN"}]

GraphData["DescartesSnark1","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

RectilinearCrossingCount[GraphData["DescartesSnark1"]]//Timing

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->86400])//AbsoluteTiming

cn=41;

(g=MinimalCrossingGraph[GraphData["DescartesSnark1"],"CrossingNumber"->cn])//PolyhedralGraphQ//Timing

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["DescartesSnark1"],"CrossingNumber"->cn,RandomSeed->n]],{n,2}],Max[OuterFace[#]]<=GraphData["DescartesSnark1","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["DescartesSnark1"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["DescartesSnark1","Circulant"]

GraphData["DescartesSnark1","Graph","Circulant"]//StyleGraphs

Complement graph

GraphData["DescartesSnark1","ComplementGraph","Name"]

Line graph

GraphData["DescartesSnark1","Line"]

LineGraphQ[GraphData["DescartesSnark1"]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "DescartesSnark1"&]

RootGraph[GraphData["DescartesSnark1"]]

Halin

GraphData["DescartesSnark1","Halin"]

GraphData["DescartesSnark1","Graph","Halin"]//StyleGraphs

Rigid

GraphData["DescartesSnark1","Rigid"]

GraphData["DescartesSnark1","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["DescartesSnark1","Flexible"]

GraphData["DescartesSnark1","Graph","Flexible"]//StyleGraphs

LCF

GraphData["DescartesSnark1","LCF"]

GraphData["DescartesSnark1","HamiltonianCycleCount"]

GraphData["DescartesSnark1","LCFSignature"]

GraphData["DescartesSnark1","LCFNotations"]

GraphData["DescartesSnark1","Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData["DescartesSnark1","UnitDistance"]

UnitDistanceGraphQ[GraphData["DescartesSnark1"],Debug->True]//Timing

GraphData["DescartesSnark1","Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["DescartesSnark1","VertexList"],GraphData["DescartesSnark1","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["DescartesSnark1","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["DescartesSnark1"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["DescartesSnark1"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"DescartesSnark1",l=FindSubgraphUnitDistanceEmbedding[GraphData["DescartesSnark1"],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData["DescartesSnark1","Matchstick"]

GraphData["DescartesSnark1","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["DescartesSnark1","Edges"]]

3D All

GraphData["DescartesSnark1","Graph",{"3D","All"}]

3D UnitDistance

GraphData["DescartesSnark1","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["DescartesSnark1","Polyhedral"]

GraphData["DescartesSnark1","Graph",{"3D","Polyhedron"}]

Properties

BetweennessCentralities

GraphData["DescartesSnark1","BetweennessCentralities"]

BetweennessCentralities[GraphData["DescartesSnark1"]]//Timing

ChordlessCycles

GraphData["DescartesSnark1","ChordlessCycles"]

ComplementChordlessCyclePolynomial

GraphData["DescartesSnark1","ComplementChordlessCyclePolynomial"]/.f_Function:>f[x]

ComplementChordlessCycles

GraphData["DescartesSnark1","ComplementChordlessCycles"]

ChordlessCyclePolynomial[GraphComplement@GraphData["DescartesSnark1"],x,Debug->True]//Timing

GraphData["DescartesSnark1","ComplementChordlessCycles"]

FindChordlessCycle[GraphComplement@GraphData["DescartesSnark1"],Infinity,All]//Timing

l={{"DescartesSnark1",{0,Missing["TooLarge",{1,2,34,3}]}}}

PropertyReplace[l,"ComplementChordlessCycles"]

RenameGraphFiles[]

CrossingNumber

GraphData["DescartesSnark1",{"RCN","CN"}]

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["DescartesSnark1"],TimeConstraint->86400])//AbsoluteTiming

CyclePolynomial

GraphData["DescartesSnark1","CyclePolynomial"]

MemoryConstrained[CyclePolynomial[,x,Debug->True],100*^9]//Timing

CyclicEdgeConnectivity

GraphData["DescartesSnark1","CyclicEdgeConnectivity"]

FindCyclicEdgeCut[g,Debug->True]//Timing

DominationNumber

GraphData["DescartesSnark1","DominationNumber"]

EdgeBetweennessCentralities

GraphData["DescartesSnark1","EdgeBetweennessCentralities"]

EdgeBetweennessCentralities[GraphData["DescartesSnark1"]]//Timing

HamiltonianCycles

GraphData["DescartesSnark1","HamiltonianCycles"]

(hc=FindHamiltonianCycle[GraphData["DescartesSnark1"]])//Timing

HamiltonianPaths

GraphData["DescartesSnark1","HamiltonianPaths"]

(hp=FindHamiltonianPath[GraphData["DescartesSnark1"]])//Timing

Hypohamiltonian

GraphData["DescartesSnark1","Hypohamiltonian"]

HypohamiltonianGraphQ[GraphData["DescartesSnark1"]]//Timing

IndependenceNumber

GraphData["DescartesSnark1","IndependenceNumber"]

LowerIndependenceNumber

GraphData["DescartesSnark1","LowerIndependenceNumber"]

LowerIndependenceNumber[GraphData["DescartesSnark1"],Method->"MOSEK",Debug->True]//Timing

MaximumIndependentVertexSets

GraphData["DescartesSnark1","MaximumIndependentVertexSets"]

MyFindIndependentVertexSet[GraphData["DescartesSnark1"],Method->{"LinearOptimization","Vector","MOSEK"},Debug->True]//Timing

MinimumCyclicEdgeCuts

GraphData["DescartesSnark1","MinimumCyclicEdgeCuts"]

Length[cuts=FindCyclicEdgeCut[,All,Debug->True]]//Timing

cuts={{2,4,117,121},{2,12,52,68},{4,12,26,28},{26,271,276,296},{28,216,260,296},{52,64,82,84},{64,68,163,165},{82,89,273,276},{84,89,264,267},{111,117,189,192},{111,121,262,267},{163,185,216,218},{165,185,187,189},{187,192,271,273},{218,260,262,264}};

HighlightGraph[GraphData["DescartesSnark1"],Style[GraphData["DescartesSnark1","Edges"][[#]],Directive[Thickness[.03],Red]]]&/@cuts

l={{"DescartesSnark1",{0,{{2,4,117,121},{2,12,52,68},{4,12,26,28},{26,271,276,296},{28,216,260,296},{52,64,82,84},{64,68,163,165},{82,89,273,276},{84,89,264,267},{111,117,189,192},{111,121,262,267},{163,185,216,218},{165,185,187,189},{187,192,271,273},{218,260,262,264}}}}};

MinimumDominatingSets

GraphData["DescartesSnark1","MinimumDominatingSets"]

FindDominatingSet[GraphData["DescartesSnark1"],Method->"MOSEK",Debug->True]//Timing

MinimumVertexCuts

GraphData["DescartesSnark1","MinimumVertexCuts"]

(cuts=FindMinimumVertexCut[GraphData["DescartesSnark1"],All,Debug->True])//Timing

l={{"DescartesSnark1",{0,cuts}}}

PropertyReplace[l,"MinimumVertexCuts"]

RenameGraphFiles[]

l={{"DescartesSnark1",{0,Length@cuts}}}

PropertyReplace[l,"MinimumVertexCutCount"]

RenameGraphFiles[]

MinimumWeightFractionalColoring

GraphData["DescartesSnark1","MinimumWeightFractionalColoring"]

FindFractionalColoring[GraphData["DescartesSnark1"],Debug->True,Method->"Dynamic"]//Timing

MinimumTotalDominatingSets

GraphData["DescartesSnark1","MinimumTotalDominatingSets"]

%[[2]]//Length

FindTotalDominatingSet[GraphData["DescartesSnark1"],Method->"MOSEK",Debug->True]//Timing

TotalDominationNumber

GraphData["DescartesSnark1","TotalDominationNumber"]

Sources

Holton and Sheehan

pp. 93–97

Wikipedia

g=Import["https://upload.wikimedia.org/wikipedia/commons/2/27/Descartes_snark.png"]

Wikipedia

https://en.wikipedia.org/wiki/Descartes_snark

In the mathematical field of graph theory, a Descartes snark is an undirected graph with 210 vertices and 315 edges. It is a snark, first discovered by William Tutte in 1948 under the pseudonym Blanche Descartes.

A Descartes snark is obtained from the Petersen graph by replacing each vertex with a nonagon and each edge with a particular graph closely related to the Petersen graph. Because there are multiple ways to perform this procedure, there are multiple Descartes snarks.

Import["https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Descartes_snark.png/200px-Descartes_snark.png"]

Portuguese Wiki site

https://pt.frwiki.wiki/wiki/Snark_de_Descartes

Import["https://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Blanche_Descartes_snark.svg/langfr-280px-Blanche_Descartes_snark.svg.png"]

GraphData["P"]

Construction

g=Import["https://upload.wikimedia.org/wikipedia/commons/2/27/Descartes_snark.png"]

Manual

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v0=Delete[v0,{197}];

AppendTo[v0,v0[[208]]+v0[[208]]-v0[[207]]]

v=(#-Mean[v0])&/@v0/500;

v

Max[v]

Length[v]

g=RotateGraph[Graph[Range@Length@v,Sort[Sort/@(UndirectedEdge@@@{e[16,194,186,193,138,137,136,102,101,100,41,32,31,5,13,16,15,14,210,208,207,196,195,194],e[193,192,191,167,166,165,140,139,138],e[136,135,134,117,116,115,103,99,102],e[100,93,106,37,28,29,36,40,41],e[31,35,34,1,19,2,3,4,5],e[4,8,11,15],e[195,187,184,192],e[135,131,130,139],e[93,95,98,99],e[40,26,24,35],e[3,9,6,14],e[196,189,190,191],e[140,133,127,134],e[103,104,105,106],e[36,42,33,34],e[59,51,52,53,54,89,88,171,170,169,168,176,155,156,76,75,74,73,72,77,78,121,120,119,118,124,148,147,177,209,197,199,182,60,63,59,58,57,56,55,54],e[121,122,123,126,125,124],e[176,175,174,173,172,171],e[72,71,70,69,68,76],e[182,181,180,179,178,177],e[58,64,65,181],e[178,146,143,125],e[68,152,151,175],e[55,85,86,172],e[71,82,81,122],e[69,149,174],e[173,92,56],e[70,84,123],e[8,7,12,10,11],e[187,183,185,188,184],e[26,30,25,27,24],e[130,128,129,132,131],e[98,96,94,97,95],e[44,47,50,45,48,44],e[29,43,51],e[37,46,49,53],e[120,107,115],e[118,113,114,117],e[1,39,38,73],e[2,23,75],e[210,203,201,199],e[207,206,209],e[168,163,162,167],e[170,164,165],e[57,67,180],e[179,141,126],{25,32},{30,33},{27,42},{101,94},{96,105},{97,104},{127,128},{132,133},{129,137},{12,13},{9,10},{6,7},{185,186},{188,189},{183,190},e[149,150,153,156],e[150,154,155],{152,154},{153,151},e[197,198,204,208],e[206,205,202,203],e[205,200,201],{200,204},{198,202},e[19,18,21,74],e[169,159,158,166],{43,44},{46,47},{48,49},{28,45},{50,52},e[119,110,109,116],e[39,17,22,23],e[38,20,22],{20,18},{17,21},e[164,157,160,163],e[157,161,162],{161,159},{158,160},e[67,66,61,60],e[63,62,66],{61,64},{62,65},e[84,83,80,77],e[78,79,83],{80,81},{79,82},e[141,142,145,147],e[142,144,148],{145,143},{144,146},e[92,91,90,89],e[91,87,88],{85,87},{90,86},e[107,108,112,113],e[108,111,114],{110,111},{109,112}})],VertexCoordinates->v,VertexLabels->Automatic,VertexSize->.2],Pi]

UnlabelGraph@g

g=

EdgeCount[g]

RecognizeGraph@g

EdgeChromaticNumber[g]

GraphGirth[g]

MyEdgeChromaticNumber[g,Debug->True]

SnarkQ[g,Debug->True]

WeakSnarkQ[g,Debug->True]//Timing

FindCyclicEdgeCut[g,Debug->True]//Timing

FindImageShapes [fail]

points=FindImageShapes[g,"Circle"]

HighlightImage[g,points]

GraphData

GraphDataClassString

GraphDataClassString["DescartesSnark1"]

GraphDataString

<<MathWorld`Graphs`

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

g=;

WriteString[file,GraphDataString[g,
"StandardName"->{"DescartesSnark",1},
"Name"->"first Descartes snark",
"Classes"->{"Snark","WeakSnark"},
"Information"->"DescartesSnarks",
"WikidataID"->"Q5263683",
"MinimumCyclicEdgeCut"->{1,4,21,26},
TimeConstraint->10,
"Skip"->{"ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","DominationNumber","LowerIndependenceNumber","LowerMatchingNumber","MinimumDominatingSet","MinimumTotalDominatingSet","TuttePolynomial"}
]]

Close[file]