Rectilinear Local Crossing Number

Author

Eric W. Weisstein
September 18, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

GraphDataCountTable

GraphDataCountTable["RectilinearLocalCrossingNumber",{"Complete"->374243}]

Parametrized families

<<MathWorld`Graphs`

Alkane

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

AlternatingGroupGraph

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

Andrasfai

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

Antelope n×n

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

Antiprism

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

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

Apollonian

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

Barbell

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

Bishop n×n

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

BlackBishop n×n

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

Book

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

Bruhat

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

Camel n×n

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

Centipede

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

CocktailParty

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

Complete

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

A374243 (except for n==8 and 14)

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

Table[Piecewise[{{4,n==8},{15,n==14}},Ceiling[1/2 (n-3-Ceiling[(n-3)/3]) Ceiling[(n-3)/3]]],{n,30}]==seq

Table[GraphData[{"Complete",n},"LocalCrossingNumber"],{n,30}]

https://doi.org/10.1016/j.jcta.2017.04.003

Table[Ceiling[(1/2)(n-3-Ceiling[((n-3)/3)])Ceiling[((n-3)/3)]],{n,20}]

lcr[n_]:=Ceiling[(1/2)(n-3-Ceiling[((n-3)/3)])Ceiling[((n-3)/3)]]

TextGrid[Table[{n,lcr[n],Piecewise[{{4,n==8},{15,n==14}}]},{n,{8,14}}],Dividers->All,Alignment->Decimal]

seq=Table[lcr[n],{n,60}]

ToString[seq]//StringLength

FindSequenceFunction[seq,n]//Head//Timing

gf=x FindGeneratingFunction[seq,x,ValidationLength->2]

StringReplace[ToString[gf,InputForm]," "->""]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0&&n∈Integers]

Table[(1/108) (89-9 (((-1))^n)+6 n (-11+2 n)+9 Cos[((n π)/3)]+(19-6 n) Cos[((2 n π)/3)]-9 (Sqrt[3]) Sin[((n π)/3)]+(Sqrt[3]) (-21+2 n) Sin[((2 n π)/3)]),{n,60}]==seq

CoefficientList[Series[(((x^4) (-1-(x^2)-(x^4)-(x^5)))/((((-1+x))^3) (((1+x+(x^2)))^2) (1+(x^3)))),{x,0,20}],x]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length@coef]

LinearRecurrence[{1,0,1,-1,0,1,-1,0,-1,1},{0,0,0,0,1,1,2,3,4,6},20]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

FullSimplify[RSolve[Prepend[Table[p[k,x]==seq[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n>0&&n∈Integers]

CompleteBipartite n×n

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

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

CompleteBipartite m×n

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

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

CompleteTripartite n×n×n

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

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

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"RectilinearLocalCrossingNumber"],
3Max[k,m,n]/2
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CrossedPrism

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

Crown

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

CubeConnectedCycle

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

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

Cycle

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

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

CycleComplement

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

DiagonalIntersection

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

Dipyramid

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

DorogovtsevGoltsevMendes

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

DoubleCone

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

Empty

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

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

FibonacciCube

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

Fiveleaper n×n

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

Flower

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

FoldedCube

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

Gear

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

Giraffe n×n

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

Goldberg

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

Grid n×n

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

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

Grid m×n

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

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

Grid n×n×n

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

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

Grid k×m×n

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

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

HalvedCube

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

Hanoi

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

Helm

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

Hypercube

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

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

Keller

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

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

King n×n

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

King m×n

Entity["Graph",{"King",{m,n}}]["RectilinearLocalCrossingNumber"]

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

Knight n×n

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

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

Knight m×n

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

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

Ladder

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

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

LadderRung

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

LindgrenSousselier

seq=Table[GraphData[{"LindgrenSousselier",6n+4},"RectilinearLocalCrossingNumber"],{n,20}]

LucasCube

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

MengerSponge

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

MoebiusLadder

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

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

Mycielski

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

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

Odd

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

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

Pan

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

Path

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

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

PathComplement

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

Pell

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

PlummerToft

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

Prism

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

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

Queen n×n

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

Rook n×n

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

RookComplement n×n

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

SierpinskiCarpet

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

SierpinskiGasket

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

SierpinskiTetrahedron

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

Spoke n×n

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

StackedBook

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

StackedPrism

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

Star

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

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

Sun

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

Sunlet

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

Tetrahedral

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

TorusGrid n×n

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

Transposition

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

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

Trapezohedron

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

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

Triangular

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

TriangularGrid

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

TriangularHoneycombAcuteKnight

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

TriangularHoneycombBishop

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

TriangularHoneycombObtuseKnight

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

TriangularHoneycombQueen

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

TriangularHoneycombRook

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

TriangularSnake

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

Web

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

Wheel

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

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

WheelComplement

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

WhiteBishop n×n

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

Zebra n×n

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