Radio Labeling

Author

Eric W. Weisstein
July 28, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Sources

http://www.idr.iitkgp.ac.in/jspui/bitstream/123456789/901/4/NB14314_Introduction.pdf

Counts

Totals of radio number labeling counts on n nodes

A340645

Table[Total[GraphData[#,"RadioLabelingCount"]&/@GraphData["Connected",n]],{n,9}]

MapIndexed[(#/._Missing:>#2[[1]])&,%]

Minimum

Table[Min[GraphData[#,"RadioLabelingCount"]&/@GraphData["Connected",n]],{n,9}]

Maximum

Table[Max[GraphData[#,"RadioLabelingCount"]&/@GraphData["Connected",n]],{n,9}]

Parametrized families table

GraphDataCountTable["RadioLabelingCount",{Sequence@@Thread[{"Complete","CompleteBipartite","CompleteTripartite"}->000012],"Dipyramid"->242522,"Empty"->000007,"PathComplement"->179184,"Star"->000012,"Wheel"->242522}]

Parametrized families

Andrasfai

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

Antelope m×n

DeleteCases[Sort[Flatten[Table[{{m,n},XXX,GraphData[{"Antelope",{m,n}},"RadioLabelingCount"]},{n,2,10},{m,2,n}],1]],{_,x_,x_}]

GraphData[{"Empty",27},"StandardNames"]

Antelope n×n

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

Antiprism

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

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

FindSequenceFunction[DeleteMissing@seq,n-2]//Factor

Apollonian

seq=Table[GraphData[{"Apollonian",n},"RadioLabelingCount"],{n,6}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Barbell

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

Bishop m×n

DeleteCases[Sort[Flatten[Table[{{m,n},0,GraphData[{"Bishop",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

Bishop n×n

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

BlackBishop m×n

DeleteCases[Sort[Flatten[Table[{{m,n},Piecewise[{
{0,Min[m,n]==1||m==n==2},
{1,m==n==3},
{4,Min[m,n]==3&&Max[m,n]==4},
{Mod[m,2]+1,m==n},
{Max[m,n]-2,Min[m,n]==2},
{n+1,n==m+1},
{Min[m,n],Mod[Min[m,n],2]==0}
},XXX],GraphData[{"BlackBishop",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

BlackBishop n×n

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

Book

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

Bruhat

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

Table[n!,{n,6}]

Centipede

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

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"RadioLabelingCount"],{n,3,20}]

Complete

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

seq=Table[GraphData[{"Complete",n},"RadioLabelingCount"],{n,20}]

CompleteBipartite m×n

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

Sort@DeleteCases[Flatten[Table[{{m,n},1,GraphData[{"CompleteBipartite",{m,n}},"RadioLabelingCount"]},{n,20},{m,n}],1],{_,x_,x_}]

Infer:

GraphData[{"CompleteBipartite",{2,17}},"Graph","Labeled"]

FindVertexCut[GraphData[{"CompleteBipartite",{2,17}}]]

CompleteBipartite n×n

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

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

Infer:

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[
{{k,m,n},
Piecewise[{{1,k==m==n},{2,k==m||m==n||n==k}},3],
GraphData[{"CompleteTripartite",{k,m,n}},"RadioLabelingCount"]
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

Inferred:

CompleteTripartite n×n×n

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

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

Infer:

CrossedPrism

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

Crown

seq=Table[GraphData[{"Crown",n},"RadioLabelingCount"],{n,3,20}]

CubeConnectedCycle

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

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

Cycle

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

seq=Table[GraphData[{"Cycle",n},"RadioLabelingCount"],{n,3,20}]

CycleComplement

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

With[{gn="CycleComplement"},
Do[
Print[{{gn,n},Length[lab=RadioNumberLabelings[GraphData[{gn,n}]]]//Timing,{lab,ByteCount[lab]}}],
{n,3,16}]];

DiagonalIntersection

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

Dipyramid

A242522

Computed for n=4 to 9; inferred for n = 10 and up

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

Inferred beyond these:

Empty

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

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

FibonacciCube

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

Fiveleaper m×n

DeleteCases[Sort[Flatten[Table[{{m,n},0,GraphData[{"Fiveleaper",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

Fiveleaper n×n

seq=Table[GraphData[{"Fiveleaper",{n,n}},"RadioLabelingCount"],{n,10}]

FoldedCube

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

Gear

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

Grid m×n

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

DeleteCases[Flatten[Table[{{m,n},Piecewise[{{0,m==n==1},{Max[m,n]-2,Min[m,n]==1},{3n-4,Min[m,n]==2}},4],GraphData[{"Grid",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1],{_,x_,x_}]

seq=Table[GraphData[{"Grid",{2,n}},"RadioLabelingCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]//Factor

Grid n×n

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

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

Grid k×m×n

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

DeleteCases[Flatten[Table[{{k,m,n},
Piecewise[{
{0,k==m==n==1||k m n==2},
{2,k m n==4},
{3,k m n==5},
{1,k m n==Max[k,m,n]},
{5,Min[k,m,n]==1&&Max[k,m,n]==3&&k m n==6},
{8,Min[k,m,n]>1||(Min[k,m,n]==1&&Max[k,m,n]==4&&k m n==8)},
{11,Min[k,m,n]==1&&Max[k,m,n]==5&&k m n==10}},4],GraphData[{"Grid",{k,m,n}},"RadioLabelingCount"]},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

Grid n×n×n

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

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

HalvedCube

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

Hanoi

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

FindSequenceFunction[DeleteMissing@seq,n]

Helm

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

HexagonalGrid n×n×n

seq=Table[GraphData[{"HexagonalGrid",{n,n,n}},"RadioLabelingCount"],{n,5}]

Hypercube

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

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

Keller

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

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

King m×n

DeleteCases[Sort[Flatten[Table[{{m,n},Piecewise[{
{0,m n≤2||m==n==2},
{1,m n==3||(Min[m,n]==2&&Max[m,n]==3)},
{6,m==n==3},
{Max[m,n]+2,Min[m,n]==3},
{Max[m,n]-2,Min[m,n]<3}},4],GraphData[{"King",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

King n×n

seq=Table[GraphData[{"King",{n,n}},"RadioLabelingCount"],{n,10}]

Knight m×n

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

DeleteCases[Sort[Flatten[Table[{{m,n},Piecewise[XXX,4],GraphData[{"Knight",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

Knight n×n

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

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

Ladder

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

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

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-1],n∈Integers]

LadderRung

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

MengerSponge

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

MoebiusLadder

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

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

Mycielski

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

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

Odd

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

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

Pan

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

Path

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

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

PathComplement

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

Prism

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

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

Queen m×n

DeleteCases[Sort[Flatten[Table[{
{m,n},
Piecewise[XXX,XXX],
GraphData[{"Queen",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"RadioLabelingCount"],{n,10}]

Rook m×n

DeleteCases[Sort[Flatten[Table[{{m,n},Piecewise[{{0,Min[m,n]==1},{(2^n)-2,Min[m,n]==2}},m n],GraphData[{"Rook",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"RadioLabelingCount"],{n,10}]

RookComplement m×n

DeleteCases[Sort[Flatten[Table[{{m,n},Piecewise[{{0,Min[m,n]==1||m==n==2},{9,Min[m,n]==2&&Max[m,n]==3}},Max[m,n]Min[m,n]],GraphData[{"RookComplement",{m,n}},"RadioLabelingCount"]},{n,10},{m,n}],1]],{_,x_,x_}]

RookComplement n×n

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

SierpinskiCarpet

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

SierpinskiSieve

seq=Table[GraphData[{"SierpinskiSieve",n},"RadioLabelingCount"],{n,5}]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"RadioLabelingCount"],{n,4}]

StackedBook n×n

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

StackedPrism n×n

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

Star

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

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

Sun

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

Sunlet

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

Tetrahedral

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

TorusGrid m×n

DeleteCases[Sort[Flatten[Table[{{m,n},m n,GraphData[{"TorusGrid",{m,n}},"RadioLabelingCount"]},{n,3,10},{m,3,n}],1]],{_,x_,x_}]

TorusGrid n×n

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

Transposition

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

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

Inferred beyond these:

Triangular

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

Inferred beyond these:

TriangularGrid

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

TriangularHoneycombAcuteKnight

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

TriangularHoneycombBishop

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

TriangularHoneycombKing

seq=Table[GraphData[{"TriangularHoneycombKing",n},"RadioLabelingCount"],{n,11}]

TriangularHoneycombObtuseKnight

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

TriangularHoneycombQueen

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

TriangularHoneycombRook

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

TriangularSnake

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

Web

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

Wheel

A242522

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

Computed for n=4 to 11; inferred for n = 12 and up

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

Inferred beyond these:

WhiteBishop m×n

DeleteCases[Sort[Flatten[Table[{{m,n},XXX,GraphData[{"WhiteBishop",{m,n}},"RadioLabelingCount"]},{n,2,10},{m,2,n}],1]],{_,x_,x_}]

WhiteBishop n×n

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