Tetrahedron-Free Graph

Author

Eric W. Weisstein
December 19, 2025

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/Tetrahedron-FreeGraph.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Tetrahedron-FreeGraph.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Nice

3D UD Tetrahedron-free with large chromatic number

TextGrid[Take[Cases[DeleteCases[ReverseSortBy[GraphData[#,{"ChromaticNumber","V","TetrahedronCount","#"}]&/@Select[GraphData["Connected"],GraphData[#,"Dimension"]==3&],{#[[1]],-#[[2]]}&],{_Missing,__}],{_,_,0,_}],10],Dividers->All,Alignment->{{{Decimal},Left}}]

Parametrized families

<<MathWorld`Graphs`

Alkane

seq=Table[GraphData[{"Alkane",n},"TetrahedronFree"],{n,10}]

AlternatingGroupGraph

seq=Table[GraphData[{"AlternatingGroupGraph",n},"TetrahedronFree"],{n,2,6}]

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"TetrahedronFree"],{n,20}]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"TetrahedronFree"],{n,20}]

Antiprism

Entity["Graph",{"Antiprism",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Antiprism",n},"TetrahedronFree"],{n,3,25}]

Apollonian

seq=Table[GraphData[{"Apollonian",n},"TetrahedronFree"],{n,5}]

Barbell

seq=Table[GraphData[{"Barbell",n},"TetrahedronFree"],{n,3,20}]

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"TetrahedronFree"],{n,20}]

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"TetrahedronFree"],{n,20}]

Book

seq=Table[GraphData[{"Book",n},"TetrahedronFree"],{n,20}]

Bruhat

seq=Table[GraphData[{"Bruhat",n},"TetrahedronFree"],{n,6}]

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"TetrahedronFree"],{n,20}]

Cameron

seq=Table[GraphData[{"Cameron",n},"TetrahedronFree"],{n,20}]

Centipede

seq=Table[GraphData[{"Centipede",n},"TetrahedronFree"],{n,10}]

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"TetrahedronFree"],{n,20}]

Complete

Entity["Graph",{"Complete",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Complete",n},"TetrahedronFree"],{n,30}]

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["TetrahedronFree"]

seq=Table[GraphData[{"CompleteBipartite",{n,n}},"TetrahedronFree"],{n,20}]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["TetrahedronFree"]

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"TetrahedronFree"],
True
},{n,10},{m,n}],1],{_,x_,x_}]

CompleteTripartite n×n×n

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["TetrahedronFree"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"TetrahedronFree"],{n,10}]

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["TetrahedronFree"]

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"TetrahedronFree"],
True
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CossidentePenttila

GraphData["CossidentePenttila","TetrahedronFree"]

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"TetrahedronFree"],{n,2,12}]

Crown

seq=Table[GraphData[{"Crown",n},"TetrahedronFree"],{n,20}]

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["TetrahedronFree"]

seq=Table[GraphData[{"CubeConnectedCycle",n},"TetrahedronFree"],{n,3,8}]

Cycle

Entity["Graph",{"Cycle",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Cycle",n},"TetrahedronFree"],{n,3,30}]

CycleComplement

seq=Table[GraphData[{"CycleComplement",n},"TetrahedronFree"],{n,3,20}]

DiagonalIntersection

seq=Table[GraphData[{"DiagonalIntersection",n},"TetrahedronFree"],{n,3,14}]

Dipyramid

seq=Table[GraphData[{"Dipyramid",n},"TetrahedronFree"],{n,3,20}]

DorogovtsevGoltsevMendes

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TetrahedronFree"],{n,0,7}]

DoubleCone

seq=Table[GraphData[{"DoubleCone",n},"TetrahedronFree"],{n,3,20}]

Empty

Entity["Graph",{"Empty",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Empty",n},"TetrahedronFree"],{n,30}]

FibonacciCube

seq=Table[GraphData[{"FibonacciCube",n},"TetrahedronFree"],{n,15}]

Fiveleaper n×n

seq=Table[GraphData[{"Fiveleaper",{n,n}},"TetrahedronFree"],{n,20}]

Flower

seq=Table[GraphData[{"Flower",n},"TetrahedronFree"],{n,5,20}]

FoldedCube

seq=Table[GraphData[{"FoldedCube",n},"TetrahedronFree"],{n,2,10}]

Gear

seq=Table[GraphData[{"Gear",n},"TetrahedronFree"],{n,3,20}]

Giraffe n×n

seq=Table[GraphData[{"Giraffe",{n,n}},"TetrahedronFree"],{n,20}]

Goldberg

seq=Table[GraphData[{"Goldberg",n},"TetrahedronFree"],{n,3,20}]

Grid n×n

Entity["Graph",{"Grid",{n,n}}]["TetrahedronFree"]

seq=Table[GraphData[{"Grid",{n,n}},"TetrahedronFree"],{n,20}]

Grid m×n

Entity["Graph",{"Grid",{m,n}}]["TetrahedronFree"]

Sort@DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"TetrahedronFree"],
True
},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n×n

Entity["Graph",{"Grid",{n,n,n}}]["TetrahedronFree"]

seq=Table[GraphData[{"Grid",{n,n,n}},"TetrahedronFree"],{n,10}]

Grid k×m×n

Entity["Graph",{"Grid",{k,m,n}}]["TetrahedronFree"]

Sort@DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"TetrahedronFree"],
True
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

HalvedCube

seq=Table[GraphData[{"HalvedCube",n},"TetrahedronFree"],{n,10}]

Hanoi

seq=Table[GraphData[{"Hanoi",n},"TetrahedronFree"],{n,6}]

Harary

GraphData["Harary","TetrahedronFree"]

Helm

seq=Table[GraphData[{"Helm",n},"TetrahedronFree"],{n,3,20}]

Hypercube

Entity["Graph",{"Hypercube",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Hypercube",n},"TetrahedronFree"],{n,0,10}]

KC

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"KC",{m,n}},"TetrahedronFree"],
XXX
},{n,10},{m,10}],1],{_,x_,x_}]

Keller

Entity["Graph",{"Keller",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Keller",n},"TetrahedronFree"],{n,5}]

King n×n

seq=Table[GraphData[{"King",{n,n}},"TetrahedronFree"],{n,20}]

King m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"King",{m,n}},"TetrahedronFree"],
Piecewise[{{True,Min[m,n]==1}},False]
},{n,10},{m,n}],1],{_,x_,x_}]

Knight n×n

Entity["Graph",{"Knight",{n,n}}]["TetrahedronFree"]

seq=Table[GraphData[{"Knight",{n,n}},"TetrahedronFree"],{n,20}]

Knight m×n

Entity["Graph",{"Knight",{m,n}}]["TetrahedronFree"]

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Knight",{m,n}},"TetrahedronFree"],
True
},{n,10},{m,n}],1],{_,x_,x_}]

KP

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"KP",{m,n}},"TetrahedronFree"],
XXX
},{n,10},{m,10}],1],{_,x_,x_}]

Ladder

Entity["Graph",{"Ladder",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Ladder",n},"TetrahedronFree"],{n,20}]

LadderRung

seq=Table[GraphData[{"LadderRung",n},"TetrahedronFree"],{n,20}]

LindgrenSousselier

seq=Table[GraphData[{"LindgrenSousselier",n},"TetrahedronFree"],{n,11}]

LucasCube

seq=Table[GraphData[{"LucasCube",n},"TetrahedronFree"],{n,15}]

MengerSponge

seq=Table[GraphData[{"MengerSponge",n},"TetrahedronFree"],{n,3}]

MoebiusLadder

Entity["Graph",{"MoebiusLadder",n}]["TetrahedronFree"]

seq=Table[GraphData[{"MoebiusLadder",n},"TetrahedronFree"],{n,3,25}]

Mycielski

Entity["Graph",{"Mycielski",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Mycielski",n},"TetrahedronFree"],{n,11}]

Odd

Entity["Graph",{"Odd",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Odd",n},"TetrahedronFree"],{n,7}]

Paley

seq=Table[GraphData[{"Paley",n},"TetrahedronFree"],{n,XXX}]

Pan

seq=Table[GraphData[{"Pan",n},"TetrahedronFree"],{n,3,20}]

Path

Entity["Graph",{"Path",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Path",n},"TetrahedronFree"],{n,20}]

PathComplement

seq=Table[GraphData[{"PathComplement",n},"TetrahedronFree"],{n,20}]

Peisert

GraphData["Peisert","TetrahedronFree"]

Pell

seq=Table[GraphData[{"Pell",n},"TetrahedronFree"],{n,9}]

PlummerToft

seq=Table[GraphData[{"PlummerToft",n},"TetrahedronFree"],{n,20}]

Prism

Entity["Graph",{"Prism",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Prism",n},"TetrahedronFree"],{n,3,25}]

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"TetrahedronFree"],{n,20}]

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"TetrahedronFree"],{n,20}]

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"TetrahedronFree"],{n,20}]

SierpinskiCarpet

seq=Table[GraphData[{"SierpinskiCarpet",n},"TetrahedronFree"],{n,4}]

SierpinskiGasket

seq=Table[GraphData[{"SierpinskiGasket",n},"TetrahedronFree"],{n,7}]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"TetrahedronFree"],{n,6}]

Spoke n×n

seq=Table[GraphData[{"Spoke",{n,n}},"TetrahedronFree"],{n,5}]

StackedBook m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedBook",{m,n}},"TetrahedronFree"],
XXX
},{n,10},{m,n}],1],{_,x_,x_}]

StackedBook n×n

seq=Table[GraphData[{"StackedBook",{n,n}},"TetrahedronFree"],{n,10}]

StackedPrism m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedPrism",{m,n}},"TetrahedronFree"],
XXX
},{m,3,8},{n,5}],1],{_,x_,x_}]

StackedPrism n×n

seq=Table[GraphData[{"StackedPrism",{n,n}},"TetrahedronFree"],{n,3,5}]

Star

Entity["Graph",{"Star",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Star",n},"TetrahedronFree"],{n,25}]

Sun

seq=Table[GraphData[{"Sun",n},"TetrahedronFree"],{n,3,20}]

Sunlet

seq=Table[GraphData[{"Sunlet",n},"TetrahedronFree"],{n,3,20}]

Tetrahedral

seq=Table[GraphData[{"Tetrahedral",n},"TetrahedronFree"],{n,6,10}]

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"TetrahedronFree"],{n,3,20}]

Transposition

Entity["Graph",{"Transposition",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Transposition",n},"TetrahedronFree"],{n,6}]

Trapezohedron

Entity["Graph",{"Trapezohedron",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Trapezohedron",n},"TetrahedronFree"],{n,3,20}]

Triangular

seq=Table[GraphData[{"Triangular",n},"TetrahedronFree"],{n,2,20}]

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"TetrahedronFree"],{n,0,20}]

TriangularHoneycombAcuteKnight

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"TetrahedronFree"],{n,20}]

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"TetrahedronFree"],{n,20}]

TriangularHoneycombObtuseKnight

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"TetrahedronFree"],{n,20}]

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"TetrahedronFree"],{n,20}]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"TetrahedronFree"],{n,20}]

TriangularSnake

seq=Table[GraphData[{"TriangularSnake",n},"TetrahedronFree"],{n,1,20,2}]

Turan

GraphData["Turan","TetrahedronFree"]

Web

seq=Table[GraphData[{"Web",n},"TetrahedronFree"],{n,3,20}]

Wheel

Entity["Graph",{"Wheel",n}]["TetrahedronFree"]

seq=Table[GraphData[{"Wheel",n},"TetrahedronFree"],{n,4,20}]

WheelComplement

seq=Table[GraphData[{"WheelComplement",n},"TetrahedronFree"],{n,4,20}]

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"TetrahedronFree"],{n,2,20}]

WhiteCamel n×n

seq=Table[GraphData[{"Bishop",{n,n}},"TetrahedronFree"],{n,{5}}]

Zebra n×n

seq=Table[GraphData[{"Zebra",{n,n}},"TetrahedronFree"],{n,20}]

Algorithms

GraphData["Q4","CliquePolynomial"]

FindClique finds *maximal* cliques.  There is no built-in WL function to find *maximum* cliques it seems;

DeleteCases[
{
#,
GraphData[#,"TetrahedronFree"],
FindClique[GraphData[#],{4}]==={}
}&/@GraphData[;;5],
{_,x_,x_}
]

FindClique[CompleteGraph[6],{4}]

GraphData["PentatopeGraph","CliquePolynomial"][x]//Expand

Coefficient[GraphData["PentatopeGraph","CliquePolynomial"][x],x,4]

GraphData["PentatopeGraph","MaximalCliquePolynomial"][x]