Hamiltonian Number

Author

Eric W. Weisstein
February 26, 2021

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

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

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

Parametrized family table

GraphDataCountTable["HamiltonianNumber",Join[{"AlternatingGroupGraph"->001710,"Andrasfai"->016789,"Antiprism"->005843,"Barbell"->005843,"BlackBishop"->087327,"Book"->047233,"Bruhat"->000142,"Centipede"->016825,"CocktailParty"->005843,"Complete"->000027,"CompleteBipartite"->005843,"CompleteTripartite"->008585,"CrossedPrism"->008586,"Crown"->005843,"CubeConnectedCycle"->036289,"Cycle"->000027,"CycleComplement"->000027,"DiagonalIntersection"->007569,"Dipyramid"->000027,"FoldedCube"->000079,"Gear"->005843,{"Grid",2}->137928,{"Grid",3}->245996,"HalvedCube"->000079,"Hanoi"->000244,"Helm"->016777,"Hypercube"->000079,"Keller"->000302,"Ladder"->005843,"MoebiusLadder"->005843,"Mycielski"->083329,"Odd"->001700,"Pan"->000027,"Path"->004275,"PermutationStar"->000142,"Prism"->005843,"SierpinskiCarpet"->001018,"SierpinskiSieve"->067771,"SierpinskiTetrahedron"->283070,"Star"->004275,"Sun"->005843,"Sunlet"->008585,"Tetrahedral"->000292,"Transposition"->000142,"Triangular"->000217,"TriangularGrid"->000217,"TriangularHoneycombBishop"->000217,
"TriangularHoneycombObtuseKnight"->000217,"TriangularHoneycombQueen"->000217,"TriangularSnake"->008585,"Web"->008586,"Wheel"->000027,"WhiteBishop"->007590},Thread[{"King","Queen","Rook","RookComplement","TorusGrid"}->000290]]]

Parametrized families

Andrasfai

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

FindSequenceFunction[seq,n]

Antiprism

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

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

Apollonian

Table[GraphData[{"Apollonian",n},"V"],{n,6}]

Table[GraphData[{"Apollonian",n},"Hamiltonian"],{n,6}]

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

GraphData[{"Apollonian",3},"HamiltonianWalks"]

Barbell

h (K_n)+1+h(K_n)+1

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

Table[2n+2,{n,3,20}]

GraphData[{"Barbell",4},"Graph","Labeled"]

GraphData[{"Barbell",4},"HamiltonianWalks"]

Bishop n×n

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

BlackBishop n×n

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

Table[Piecewise[{{6,n==3}},Ceiling[n^2/2]],{n,10}]

Book

A047233

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

Table[Piecewise[{{3n,Mod[n,2]==0},{3n+1,Mod[n,2]==1}}],{n,20}]

GraphData[{"Book",3},"Graph",{"UnitDistance","Labeled"}]

GraphData[{"Book",3},"HamiltonianWalks"][[-1]]

GraphData[{"Book",4},"Graph",{"UnitDistance","Labeled"}]

GraphData[{"Book",4},"HamiltonianWalks"][[1]]

Centipede

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

FindSequenceFunction[seq,n]

CocktailParty

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

Complete

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

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

CompleteBipartite m×n

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

Goodman, S.~E.\ and Hedetniemi, S.~T.  ``On Hamiltonian Walks in Graphs.''  {\it SIAM J.\ Comput.} {\bf 3}, 214--221, 1974.

DeleteCases[Sort@Flatten[Table[{
{m,n},
2Max[m,n],
GraphData[{"CompleteBipartite",{m,n}},"HamiltonianNumber"]
},{n,10},{m,n}],1],{_,x_,x_}]

Inferred from formula for these:

CompleteBipartite n×n

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

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

CompleteKPartite

Goodman, S.~E.\ and Hedetniemi, S.~T.  ``On Hamiltonian Walks in Graphs.''  {\it SIAM J.\ Comput.} {\bf 3}, 214--221, 1974.

data=DeleteCases[With[{ind=Lookup[GraphData[#,"NotationRules"],"CompleteKPartite"]},{#,If[Length[ind]==1||Total[ind]≥2Max[ind],Total[ind],2Max[ind]],GraphData[#,"HamiltonianNumber"]}]&/@GraphData["CompleteKPartite"],{_,x_,x_}]

CompleteTripartite k×m×n

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

Goodman, S.~E.\ and Hedetniemi, S.~T.  ``On Hamiltonian Walks in Graphs.''  {\it SIAM J.\ Comput.} {\bf 3}, 214--221, 1974.

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

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

DeleteCases[Sort@Flatten[Table[{
{k,m,n},
Piecewise[{{k+m+n,k+m+n≥2Max[k,m,n]}},2Max[k,m,n]],
GraphData[{"CompleteTripartite",{k,m,n}},"HamiltonianNumber"]
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

Filled in from formula for these:

CompleteTripartite n×n×n

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

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

CrossedPrism

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

FindSequenceFunction[seq,n-1]//Factor

Crown

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

FindSequenceFunction[seq,n-2]

CubeConnectedCycle

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

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

FindSequenceFunction[seq,n-2]

Table[(2^n)n,{n,3,8}]

Cycle

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

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

Empty

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

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

FibonacciCube

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

Table[GraphData[{"FibonacciCube",n},"V"],{n,15}]

Table[GraphData[{"FibonacciCube",n},"Hamiltonian"],{n,15}]

FoldedCube

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

FindSequenceFunction[seq,n-1]

Gear

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

FindSequenceFunction[DeleteMissing@seq,n-2]

Grid m×n

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

DeleteCases[Sort@Flatten[Table[{
{m,n},
Piecewise[{{1,m==n==1},{2Max[m,n]-2,m n==Max[m,n]},{m n+1,Mod[m n,2]==1}},m n],
GraphData[{"Grid",{m,n}},"HamiltonianNumber"]
},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n

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

A137928

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

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

Table[Piecewise[{{1,n==1},{n^2,Mod[n,2]==0},{n^2+1,Mod[n,2]==1}}],{n,10}]

5×5

g=GraphData[{"Grid",{5,5}},"Graph","Labeled"]

walk=FindHamiltonianWalk[g,Method->"HamiltonianPath"]

HamiltonianWalkQ[g,walk]

HamiltonianWalkHighlight[g,walk]

7×7

g=GraphData[{"Grid",{7,7}},"Graph","Labeled"]

walk=FindHamiltonianWalk[g,Method->"HamiltonianPath"]

HamiltonianWalkQ[g,walk]

HamiltonianWalkHighlight[g,walk]

Grid k×m×n

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

DeleteCases[Sort@Flatten[Table[{
{k,m,n},
Piecewise[{{1,k==m==n==1},{2Max[k,m,n]-2,k m n==Max[k,m,n]},{k m n+1,Mod[k m n,2]==1}},k m n],
GraphData[{"Grid",{k,m,n}},"HamiltonianNumber"]
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

Grid n×n×n

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

A245996

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

Table[Piecewise[{{1,n==1},{n^3,Mod[n,2]==0},{n^3+1,Mod[n,2]==1}}],{n,10}]

3×3

g=GraphData[{"Grid",{3,3,3}},"Graph","Labeled"]

walk=FindHamiltonianWalk[g,Method->"HamiltonianPath"]

HamiltonianWalkQ[g,walk]

HamiltonianWalkHighlight[g,walk]

5×5

g=GraphData[{"Grid",{5,5,5}},"Graph","Labeled"]

walk=FindHamiltonianWalk[g,Method->"HamiltonianPath"]

HamiltonianWalkQ[g,walk]

HamiltonianWalkHighlight[g,walk]

HalvedCube

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

FindSequenceFunction[seq,n]

Hanoi

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

FindSequenceFunction[seq,n]

Helm

A016777

seq=Table[GraphData[{"Helm",n},"HamiltonianNumber"],{n,3,10}]

Table[3n+1,{n,3,10}]

GraphData[{"Helm",3},"Graph","Labeled"]

GraphData[{"Helm",3},"HamiltonianWalks"]

GraphData[{"Helm",4},"Graph","Labeled"]

GraphData[{"Helm",4},"HamiltonianWalks"]

GraphData[{"Helm",5},"Graph","Labeled"]

Hypercube

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

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

KayakPaddle

DeleteCases[{#,{1,1,2}.Lookup[GraphData[#,"NotationRules"],"KayakPaddle"],GraphData[#,"HamiltonianNumber"]}&/@GraphData["KayakPaddle"],{_,x_,x_}]

walk=GraphData[{"KayakPaddle",{3,4,2}},"HamiltonianWalks"][[1]]

HamiltonianWalkHighlight[GraphData[{"KayakPaddle",{3,4,2}},"Graph","Labeled"],walk]

walk=GraphData[{"Barbell",3},"HamiltonianWalks"][[1]]

HamiltonianWalkHighlight[GraphData[{"Barbell",3},"Graph","Labeled"],walk]

Keller

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

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

King m×n

Block[{m,n},DeleteCases[{
#,
{m,n}=("King"/.GraphData[#,"NotationRules"]);Piecewise[{{1,m==n==1},{2(n-1),Min[m,n]==1}},m n],
GraphData[#,"HamiltonianNumber"]
}&/@GraphData["King"],{_,x_,x_}]]

King n×n

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

FindSequenceFunction[seq,n]

Knight m×n

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

Sort[Lookup[GraphData[#,"NotationRules"],"Knight"]&/@GraphData[{"Knight","Hamiltonian"}]]

Select[Sort[Flatten[Table[{m,n},{n,15},{m,n}],1]],HamiltonianGraphQ[KnightGraph@@#]&]

ham[m_,n_]:=m==n==1||(Mod[m n,2]==0&&((Min[m,n]==3&&Max[m,n]>9)||Min[m,n]>4))

Select[Sort[Flatten[Table[{m,n},{n,15},{m,n}],1]],ham@@#&]==%92

DeleteCases[Sort[Flatten[Table[{
{m,n},
GraphData[{"Knight",{m,n}},"HamiltonianNumber"],
Piecewise[{{1,m==n==1},{Missing["NotApplicable"],m==n==3||Min[m,n]<3},{m n,(Mod[m n,2]==0&&((Min[m,n]==3&&Max[m,n]>9)||Min[m,n]>4))}},m n+1]
},{n,10},{m,n}],1]],{_,x_,x_}]

walk=GraphData[{"Knight",{3,5}},"HamiltonianWalks"][[1]]

walk=GraphData[{"Knight",{3,5}},"HamiltonianWalkCount"]

HamiltonianWalkHighlight[GraphData[{"Knight",{3,5}},"Graph","Labeled"],walk]

FindHamiltonianWalk[GraphData[{"Knight",#}],Method->"HamiltonianPath",Debug->False]&/@{{3,4},{3,5},{3,6},{3,7},{3,8},{4,4},{4,5},{4,6},{4,7},{4,8},{4,9},{4,10}}

With[{i={3,6}},
FindHamiltonianWalk[GraphData[{"Knight",i}],Method->"ClosestNeighborsRandomized",MaxIterations->10^5,"HamiltonainNumber"->2+Times@@i,Debug->False]]//Timing

With[{i={4,4}},
FindHamiltonianWalk[GraphData[{"Knight",i}],Method->"ClosestNeighborsRandomized",MaxIterations->5*^5,"HamiltonainNumber"->2+Times@@i,Debug->False]]//Timing

With[{i={3,4}},
FindHamiltonianWalk[GraphData[{"Knight",i}],Method->"ClosestNeighborsRandomized",MaxIterations->10^4,"HamiltonainNumber"->2+Times@@i,Debug->False]]//Timing

With[{i={3,4}},
FindHamiltonianWalk[GraphData[{"Knight",i}],Method->"ClosestNeighborsRandomized",MaxIterations->10^4,"HamiltonainNumber"->3+Times@@i,Debug->False]]//Timing

Knight n×n

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

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

Table[Piecewise[{{1,n==1},{Missing["NotApplicable"],n==2||n==3},{18,n==4},{n^2,Mod[n,2]==0},{n^2+1,Mod[n,2]==1}}],{n,20}]

4×4

GraphData[{"Knight",{4,4}},"HamiltonianWalks"]

walk={1,7,9,2,8,15,6,4,6,13,11,5,3,12,14,7,16,10};

HamiltonianWalkQ[GraphData[{"Knight",{4,4}}],walk]

HamiltonianWalkHighlight[GraphData[{"Knight",{4,4}},"Graph","Labeled"],walk]

5×5

GraphData[{"Knight",{5,5}},"HamiltonianWalks"]

walk=%[[2]]

HamiltonianWalkHighlight[GraphData[{"Knight",{5,5}},"Graph","Labeled"],walk]

Ladder

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

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

LadderRung

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

Lollipop

GraphData[{"Lollipop",{6,3}},"Graph","Labeled"]

GraphData[{"Lollipop",{6,3}},"HamiltonianNumber"]

DeleteCases[Flatten[Table[{{n,k},GraphData[{"Lollipop",{n,k}},"HamiltonianNumber"],n+2k},{n,3,7},{k,1,5}],1],{_,x_,x_}]

MengerSponge

Table[GraphData[{"MengerSponge",n},"HamiltonianNumber"],{n,3}]

MoebiusLadder

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

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

Mycielski

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

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

Odd

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

A001700

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

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

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

Table[Binomial[-1+2 n,-1+n],{n,7}]

Pan

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

Path

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

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

Prism

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

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

Queen m×n

Block[{m,n},DeleteCases[{
#,
{m,n}=("Queen"/.GraphData[#,"NotationRules"]);m n,
GraphData[#,"HamiltonianNumber"]
}&/@GraphData["Queen"],{_,x_,x_}]]

Queen n×n

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

FindSequenceFunction[seq,n]

Rook m×n

Block[{m,n},DeleteCases[{
#,
{m,n}=("Rook"/.GraphData[#,"NotationRules"]);m n,
GraphData[#,"HamiltonianNumber"]
}&/@GraphData["Rook"],{_,x_,x_}]]

Rook n×n

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

FindSequenceFunction[seq,n]

RookComplement m×n

Block[{m,n},DeleteCases[{
#,
{m,n}=("RookComplement"/.GraphData[#,"NotationRules"]);Piecewise[{{1,m==n==1},{Missing["NotApplicable"],Min[m,n]==1||m==n==2}},m n],
GraphData[#,"HamiltonianNumber"]
}&/@GraphData["RookComplement"],{_,x_,x_}]]

RookComplement n×n

Table[GraphData[{"RookComplement",{n,n}},"HamiltonianNumber"],{n,10}]

SierpinskiCarpet

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

FindSequenceFunction[seq,n]

SierpinskiSieve

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

FindSequenceFunction[seq,n]//FullSimplify

SierpinskiTetrahedron

A283070

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

FindSequenceFunction[seq,n]//FullSimplify

Star

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

Table[GraphData[{"Star",n},"HamiltonianNumber"],{n,25}]

Sun

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

FindSequenceFunction[seq,n-2]//Factor

Sunlet

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

FindSequenceFunction[seq,n-2]

Tadpole

DeleteCases[{#,{1,2}.Lookup[GraphData[#,"NotationRules"],"Tadpole"],GraphData[#,"HamiltonianNumber"]}&/@GraphData["Tadpole"],{_,x_,x_}]

GraphData["Tadpole"]

walk=GraphData[{"Tadpole",{3,4}},"HamiltonianWalks"][[1]]

HamiltonianWalkHighlight[GraphData[{"Tadpole",{3,4}},"Graph","Labeled"],walk]

Tetrahedral

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

FindSequenceFunction[seq,n-5]//Factor

Binomial[n,3]

Transposition

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

Table[GraphData[{"Transposition",n},"HamiltonianNumber"],{n,5}]

Triangular

seq=Table[GraphData[{"Triangular",n},"HamiltonianNumber"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Factor

Binomial[n,2]

TriangularGrid

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

FindSequenceFunction[seq,n+1]

Binomial[n+2,2]

TriangularHoneycombAcuteKnight

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

TriangularHoneycombBishop

A000217

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

TriangularHoneycombObtuseKnight

A000217

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"HamiltonianNumber"],{n,4,20}]

TriangularHoneycombQueen

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

FindSequenceFunction[seq,n]

Binomial[n+1,2]

TriangularHoneycombRook

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

Web

A008586

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

Table[4n,{n,3,20}]

GraphData[{"Web",3},"Graph","Labeled"]

GraphData[{"Web",3},"HamiltonianWalks"]

GraphData[{"Web",4},"Graph","Labeled"]

GraphData[{"Web",4},"HamiltonianWalks"]

GraphData[{"Web",5},"Graph","Labeled"]

Wheel

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

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

WhiteBishop n×n

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

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

Checks

DeleteCases[{#,GraphData[#,"HamiltonianNumber"],HamiltonianNumber[GraphData[#]]}&/@GraphData["Connected",;;6],{_,n_,n_}]//Timing

DeleteCases[{#,GraphData[#,"HamiltonianNumber"],HamiltonianNumber[GraphData[#]]}&/@GraphData["Connected",;;7],{_,n_,n_}]//Timing

DeleteCases[{#,GraphData[#,"HamiltonianNumber"],HamiltonianNumber[GraphData[#]]}&/@GraphData["Connected",;;8],{_,n_,n_}]//Timing

DeleteCases[{#,GraphData[#,"HamiltonianNumber"],HamiltonianNumber[GraphData[#]]}&/@GraphData["Connected",;;9],{_,n_,n_}]//Timing