Intrinsically Linked Graph

Author

Eric W. Weisstein
March 20, 2023

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

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

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

Counts

<<MathWorld`Graphs`

Connected

Length/@Table[GraphData[{"Connected","IntrinsicallyLinked"},n],{n,7}]

Do[Print[{n,Length@ConnectedGraphs[n]//Timing}],{n,8}]

Do[Print[{n,Count[ConnectedGraphs[n],_?IntrinsicallyLinkedGraphQ]//Timing}],{n,8}]

All

Length/@Table[GraphData["IntrinsicallyLinked",n],{n,7}]

Known

TextGrid[Table[{n,GraphData[{"Connected","IntrinsicallyLinked"},n]},{n,10}],Dividers->All]

Missing

Select[GraphData["Connected",6],MissingQ[GraphData[#,"LinklesslyEmbeddable"]]&]

Select[GraphData["Connected",7],MissingQ[GraphData[#,"LinklesslyEmbeddable"]]&]

Select[GraphData["Connected",8],MissingQ[GraphData[#,"LinklesslyEmbeddable"]]&]

Select[GraphData["Connected",9],MissingQ[GraphData[#,"LinklesslyEmbeddable"]]&]

Select[GraphData["Connected",10],MissingQ[GraphData[#,"LinklesslyEmbeddable"]]&]

Example

2-D

V5

ShowGraphArray[{CompleteGraph[6],CompleteGraph[3,3,1]}]

V7

GraphicsGrid[{Show[GraphData[#],PlotLabel->GraphData[#,"Notation"],BaseStyle->Italic]&/@{"K6","K1,3,3"}}]

3-D

Ed Pegg
Jan. 30, 2009

The simplest lattice without 4 coplanar points is the 3x3x2 grid.

pts=Flatten[Table[{a,b,c},{a,0,2},{b,0,2},{c,0,1}],2];

Eliminating symmetries, there are 20 different 6 point sets, no 4 on a plane.

Graphics3D[{Sphere[pts[[#]],.1]&/@{1,2,5,12,16,17},Tube[#,.05]&/@(pts[[#]]&/@Subsets[{1,2,5,12,16,17},{2}])},Boxed->False,SphericalRegion->True,ViewPoint->{3,-1,6}]

GraphDataCountTable

GraphDataCountTable["IntrinsicallyLinked"]

Parametrized families

Alkane

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

AlternatingGroupGraph

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

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"IntrinsicallyLinked"],{n,10}]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"IntrinsicallyLinked"],{n,15}]

Antiprism

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

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

Apollonian

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

Barbell

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

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"IntrinsicallyLinked"],{n,10}]

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"IntrinsicallyLinked"],{n,10}]

Book

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

Bruhat

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

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"IntrinsicallyLinked"],{n,10}]

Centipede

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

CocktailParty

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

Complete

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

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

CompleteBipartite m×n

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

Table[GraphData[{"CompleteBipartite",{m,n}},"IntrinsicallyLinked"],{n,10},{m,n}]//Column

CompleteBipartite n×n

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

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

FindSequenceFunction[seq,n]//FullSimplify

CompleteTripartite k×m×n

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

DeleteCases[Sort@Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"IntrinsicallyLinked"],
Piecewise[{{False,k m n==Max[k,m,n]||(Min[k,m,n]==1&&k m n/Max[k,m,n]==2)}},True]
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

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

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

FindSequenceFunction[seq,n]//FullSimplify

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"IntrinsicallyLinked"],{n,2,10}]

Crown

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

CubeConnectedCycle

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

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

Cycle

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

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

CycleComplement

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

DiagonalIntersection

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

Dipyramid

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

DoubleCone

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

Empty

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

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

FibonacciCube

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

Flower

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

FoldedCube

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

Gear

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

Giraffe n×n

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

Goldberg

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

Grid m×n

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

Table[GraphData[{"Grid",{m,n}},"IntrinsicallyLinked"],{n,10},{m,n}]//Column

Grid n×n

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

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

Grid k×m×n

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

Table[GraphData[{"Grid",{k,m,n}},"IntrinsicallyLinked"],{n,5},{m,n},{k,m}]//Column

Tally[GraphData[{"Grid",{2,3,4}},"DistanceMatrix"]//Normal//Flatten]

Grid n×n×n

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

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

HalvedCube

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

Hanoi

seq=Table[GraphData[{"Hanoi",n},"IntrinsicallyLinked"],{n,5}]

Helm

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

Hypercube

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

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

Keller

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

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

King n×n

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

Knight m×n

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

Knight n×n

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

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

Ladder

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

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

LadderRung

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

MengerSponge

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

MoebiusLadder

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

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

Mycielski

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

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

Odd

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

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

Pan

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

Path

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

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

PathComplement

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

Prism

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

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

Queen n×n

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

Rook n×n

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

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"IntrinsicallyLinked"],{n,10}]

SierpinskiCarpet

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

SierpinskiGasket

seq=Table[GraphData[{"SierpinskiGasket",n},"IntrinsicallyLinked"],{n,5}]

SierpinskiTetrahedron

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

StackedBook

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

StackedPrism

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

Star

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

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

Sun

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

Sunlet

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

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"IntrinsicallyLinked"],{n,3,10}]

Transposition

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

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

Trapezohedron

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

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

Triangular

seq=Table[GraphData[{"Triangular",n},"IntrinsicallyLinked"],{n,2,10}]

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"IntrinsicallyLinked"],{n,0,10}]

TriangularHoneycombAcuteKnight

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

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"IntrinsicallyLinked"],{n,9}]

TriangularHoneycombObtuseKnight

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

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"IntrinsicallyLinked"],{n,10}]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"IntrinsicallyLinked"],{n,10}]

TriangularSnake

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

Web

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

Wheel

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

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

WheelComplement

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

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"IntrinsicallyLinked"],{n,2,10}]

Zebra n×n

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