Scramble Number

Author

Eric W. Weisstein
April 23, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Summary

DateString[]

table=GraphDataCountTable["ScrambleNumber"]

TextGrid[OEISUpdates[table],Dividers->All]

Classes

Complete

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

CompleteBipartite m×n

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

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

CompleteBipartite n×n

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

Table[GraphData[{"CompleteBipartite",{n,n}},"ScrambleNumber"],{n,20}]

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{{k,m,n},k+m+n-Max[k,m,n],GraphData[{"CompleteTripartite",{k,m,n}},"ScrambleNumber"]},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

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

Table[GraphData[{"CompleteTripartite",{n,n,n}},"ScrambleNumber"],{n,10}]

Cycle

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

Table[GraphData[{"Cycle",n},"ScrambleNumber"],{n,3,20}]

Grid m×n

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

Grid n×n

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

Grid k×m×n

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

https://arxiv.org/pdf/2103.15253.pdf

Select[{#,Lookup[GraphData[#,"NotationRules"],"Grid"]}&/@GraphData["Grid"],Length[#[[2]]]==3&]

Select[%,#[[2,1]]≥#[[2,2]]#[[2,3]]/2&]

Grid n×n×n

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

https://arxiv.org/pdf/2103.15253.pdf

Select[Flatten[Table[{l,m,n},{n,2,10},{m,2,n},{l,2,m}],2],#[[1]]≥#[[2]]#[[3]]/2&]

Ladder

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

Path

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

Star

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

Wheel

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

Table[GraphData[{"Wheel",n},"ScrambleNumber"],{n,3,20}]