Extremal graphs for some problems on cycles in graphs

Transcription

1 Extremal graphs for some problems on cycles in graphs Klas Markström Abstract. This paper contain a collection of extremal graphs for somequestionsoncyclesingraphs.thegraphshavebeenfoundby exhaustive computer search. I list the extremal graphs and values for the maximum and minumnumerofcyclesinagraph,graphswithoutcyclesoflength4 and8relatingtoaconjectureoferdösandgyarfasandthesmallest 3-connected non-hamiltonian cubic graphs of class I. 1. Introduction Many basic questions regarding the cycle structure of graphs in general, and cubic graphs in particular, are very poorly understood. In the literature we find unsolved problems of all degree of sophistication, from the cycle double cover conjecture to the hamiltonicity of various classes of graphs. In this paper I list extremal graphs and values for some graph properties relatedtocyclesingraphswhichihavehappendtobeworkingonduring thelastfewyears. NexttomoretraditionalpenandpaperworkIhave done some computational work which has been collected in this note. The four different problems treated are briefly: the maximum number of cycles, the minimum number of cycles, the existence of cycles of length a power of 2, and finding nonhamiltonian 3-connected, 3-edge colourable cubic graphs. 2. Graphs with many cycles In[ES81] Entringer and Slater studied graphs with the maximum possible number ofcyclesamongallgraphsonnverticesandmedges. More specificallytheydefinedψ(g)tobethenumberofcyclesinthegraphg andψ(k)asthemaximumnumberofcyclesinagraphwithn+k 1 edges.furthermoretheyshowedthatgivenanyvalueofkthereisacubic graphgon2(k 1)verticessuchthatψ(G)=ψ(k),i.e.thereisalwaysa reasonably small extremal graph for a given k. Sincekisthedimensionofthecyclespaceofagraphonn+k 1 edgeswefindthatψ(k)<2 k. EntringerandSlaterprovedthatψ(k) 1

2 2 k 1 +k 2 3k+3bycalculatingthenumberofcyclesintheMöbius wheels. Using an exhaustive computer search they found the value of ψ(k) fork 8andbasedonthesevaluesconjecturedthatψ(k) 2 k 1. I have extended the computer search for cubic graphs on 2(k 1) vertices forwhichψ(g)=ψ(k).ihavefoundtheextremalgraphsfork 11.For 12 k 22Ihavecomputedlowerboundsforψ(k)bynarrowingour searchtographswithhighgirth.theresultsaregivenintable1. In[ES81] it was conjectured that all cubic graphs which are extremal forψwouldhaveaslargegirthasispossibleforacubicgraphon2(k 1). This conjecture was disproved in[gui96], however it still seems to be true thattheextremalgraphstendtohaveagirthwhichisclosetothelargest possibleone. Thusitisnotunreasonabletoexpectthelowerboundsfor ψ(k)givenheretoactuallybethevalueofψ(k). ThegeneralisedPetersengraphGP(n,m),n 3,1 m<n/2,isa cubicgraphwithvertex-set{u i ;i Z n } {v i ;i Z n },andedge-set {u i u i+1,u i v i,v i v i+m ;i Z n }. Fork=6wehaveseenthatGP(6 1,2),theordinaryPetersengraph, isextremalwithrespecttoψ(g),likewisefork=5thegraphgp(5 1, 2), the cube, is extremal. Motivated by these to initial coincidences we computed the number of cycles in all small generalised Petersen graphs andfoundthatforsuitablemtheycomeverycloseindeedtotheextremal valueofψ(n 1),seeTable2.Wethusmakethefollowingconjecture. Conjecture2.1. Letp(k)=max m ψ(gp(k 1,m)). (1) p(k)=2 k 1 +f(k),wheref(k)isafunctionnotboundedbyany polynomial. (2) ψ(k) 2 k 1 =O(p(k) 2 k 1 ). Fromourdataforsmallkonefindadecentfitwithf(k) = O(k lnk ). However there is little else in support for a sharper conjecture. Thatp(k)isgreaterthan2 k 1 isimmediatesincegp(k 1,1)isthe familyofordinarycyclicladdersandforthesewehavethat ψ(gp(k 1,2)) 2 k 1 +(k 1)(k 2)+2. 2

3 k ψ(k) ψ(k)/2 k 1 girth Extremalgraphs /3 K 4,3-cage /4 K 3,3,4-cage /4 The Möbius ladder /5 The Petersen graph, 5-cage /5 Figure /6 The Heawood graph, 6-cage /5 Figure /6 Figure /5,6 Figure /6 Figure /7 The McGee graph, 7-cage /7 Figure /7 The Coxeter graph, snark /8 The Tutte-Coxeter graph, 8-cage /7 Figure /8 Figure /8 Figure /8 Figure /8 Figure /8 Figure 7 Table 1. Valuesandboundsforψ(k). Inthecolumn labelledgirtthefirstvalueistheminimumgirthofthe graphssearchedtogiveourboundandaftertheslash markwelistthegirthsoftheextremalgraphsfound. Figure1.Graphsmaximisingψ(G)fork=12. 3

4 k ψ(k) p(k) Table2. Valuesofp(k)fork 17comparedtoourbest lower bounds for ψ(k) Figure2. Graphsmaximisingψ(G)fork=16,18. Figure3.Graphsmaximisingψ(G)fork=20. 4

5 Figure4. Graphsgivingourvalueandlowerboundfor ψ(k)fork=22,26. Figure5. Graphsgivingourlowerboundforψ(k)for k=32,34. Figure6. Graphsgivingourlowerboundforψ(k)for k=36,38. 5

6 Figure7. Graphsgivingourlowerboundforψ(k)for k=40,42. 6

7 3. Graphs with few cycles In[BCE86]BarefootClarkandEntringerstudiedwhatismoreorlessto opposite question of the previous section, namely how many cycles a graph onnatleastmusthave.sincetreesbydefinitiondonothaveanycycleswe must restrict our graph in order to get a meaningful question. Barefoot et aldefinedf k (n)asthesmallestnumberofcyclesinanyk-connectedcubic graph on n vertices. Barefootetaldeterminedtheexactvalueoff 1 (n)andfoundthefamily ofextremalgraphssatisfyingthisbound,f 1 turnsouttobelinearinn. Theyalsofoundthevaluef 2 (n),whichisquadraticinn,anfoundthe corresponding extremal graphs. Fork=3f k (n)hasnotyetbeendetermined.in[bce86]itwasshown thatf 3 (n)isboundedfromabovebyn2 nc,c= log8 log9,anditwasconjectured thatf 3 (n)islargerthananypolynomial. Thisconjecturewasprovedby AldredandThomassen[AT97]whoshowedthat2 n0.17 <f 3 (n)forlarge n.theseboundsaretothisdatethebestfoundforf 3 (n). Obviouslynofamilyofextremalgraphsforf 3 (n)hasbeenfound,butin [BCE86]theextremalgraphsforn 14wasfoundbycomparingasimple family of planar graphs with values of ψ(g) from a computer generated table.thefamilyg n mentionedisconstructedasfollows.letg n ={K 4 }. AgraphG 2 G n+1 isconstructedfromagraphg 1 ing n byexpandinga vertexv,containedinasfewcyclesofg 1 aspossible,intoatriangle. It wasalsonotedwithoutproofthatifg G n thenψ(g)=f n/2+5 n/2+4, wheref n isthenthfibonaccinumber. Wehaveverifiedandextendedthesearchforextremalgraphsforf 3 (n) toalln 20andwefoundthatalltheextremalgraphsthusfarbelong tog n. Asnotedin[BCE86]thiscannotcontinuetoholdforalln,the Fibonacci number grow exponentially, and an interesting challenge is thus tofindthesmallestnforsuchthatf 3 (n)<f n/2+5 n/2+4. Theextremalgraphsforn 20aredisplayedinFigures8to13. 7

8 Figure8. Graphsminimisingψ(G)forn=8,10. Figure9. Graphsminimisingψ(G)forn=12. Figure10.Graphsminimisingψ(G)forn=14. 8

9 Figure11.Graphsminimisingψ(G)forn=16. Figure12.Graphsminimisingψ(G)forn=18. 9

10 Figure13.Graphsminimisingψ(G)forn=20. 10

11 n Table3. ThenumberofcubicgraphswithnoC 4 andc Graphswithfewcyclesoflength2 k In 1995 Erdös and Gyarfas made the following conjecture. Conjecture 4.1. Every graph with minimum degree at least 3 contains a cyclewhoselengthisapowerof2. Erdösoffered$100foraproofand$50foracounterexample.Apartfrom Shauger s results on claw-free graphs[sha98, DS01] there seems to be very little published on this conjecture. Assume that G is a, edge and vertex, minimal counterexample to this conjectureand thatuand v are twoverticesof G. If d(u) 3and d(v) 3then{u,v}cannotbeanedge;ifitwasthenG\{u,v}would be a counterexample with fewer edges than G. Thus a counterexample mustconsistofanindependentsetv 1 ofverticesofdegreeatleast4,and anonemptysetv 2 =V\V 1 ofverticesofdegree3. Using this observation Gordon Royle [Roy] used a modified version of Brendan McKay s graph generator makeg[mck] to generate graphs withoutc 4 sandthedescribeddegreestructure.roylegeneratedallrelevant graphs on less than 16 vertices and found no counterexamples. Inordertoextendthissearchfurtherwechoosetolookatgraphswith V 1 =,i.ecubicgraphs.weusedgunnarbrinkman scubicgraphgenerator minibaum[bri96] to generate all cubic graphs on less than 29 vertices andasimplefortranprogramtocheckfortheexistenceofcyclesoflength 4,8 and 16. No counterexamples to the conjecture was found. However on 24 vertices we found the smallest cubic graphs without cycles of lengths 4 and8.thesegraphsaredisplayedinfig.14.wenotethatthelowerright ofthefourgraphscanbeconstructedfromk 4 berepeatedlyexpanding vertices into triangles, it is also the only planar graph among the four. InTable3wedisplaythenumberofcubicgraphsonnverticeshaving nocyclesoflength4or8.fromthedistributionofshortcyclesinrandom cubicgraphs,seee.g. thenicesurveyin[wor99],wefindthatforthese smallnthenumberofgraphswithoutanycyclesoflength4and8ismuch larger than the asymptotic proportion of such graphs among the cubic graphs. 11

12 Figure 14.The cubic graphs on 24 which contains neitherac 4 norac 8 12

13 5. Nonhamiltonian cubic graphs of class I The hamiltonicity of cubic graphs is a well investigated property. It is of course easy to construct a nonhamiltonian 1-connected cubic graphs and it is only slightly harder to find a 2-connected nonhamiltonian graph. Among the 3-connected cubic graphs the smallest nonahmiltonian graph isthewellknownpetersen-graph,andfromitthereisanumberofways to construct families of larger nonhamiltonian graphs. Graphs constructed in this way are typically of class II, i.e. they are not 3-edge-colourable. AmongcubicgraphsofclassItheplanargraphshavereceivedalotof attention, first in connection with the 4-colour theorem. The first nonhamiltonian example was found by Tutte in[tut46]. Later on several examples of non-hamiltonian cubic graphs on 38 vertices were found by Lederberg, Barnette, and Bosk. Holton and McKay proved that all 3- connected planar cubic graphs on at most 36 vertices are hamiltonian, thus proving the minimality of the examples on 38 vertices. Here we have searched for 3-connected, non-planar, non-hamiltonian, cubic graphs of class I. We used a fortran program to sieve for nonhamiltonian graphs among all cubic graphs on n vertices and among those weusedmathematicatocheckwetherthefoundgraphswereclassiand 3-connected. The cubic graphs were generated using minibaum[bri96]. Alltrianglefreecubicgraphsonatmost22verticesweretested,therestriction to triangle free graphs coming from the fact that if we contract atriangleinacubicgraphwepreserveboththechromaticindexandthe hamiltonicity of the graph. InFigure15weshowthesmallestsuchgraphsandinFigure16weshow allsuchgraphson22vertices. ThefirstfourgraphsinFigure16canbe constructed from the graph in Figure 15, the fifth graph is the generalised Petersen graph GP(11, 2). 13

14 Figure 15. The 3-connected class-i nonhamiltonian cubic graph on 20 vertices. Figure 16 14

15 References [AT97] R. E. L. Aldred and Carsten Thomassen, On the number of cycles in 3- connected cubic graphs, J. Combin. Theory Ser. B 71(1997), no. 1, [BCE86] C. A. Barefoot, Lane Clark, and Roger Entringer, Cubic graphs with the minimum number of cycles, Proceedings of the seventeenth Southeastern international conference on combinatorics, graph theory, and computing (Boca Raton, Fla., 1986), vol. 53, 1986, pp [Bri96] Gunnar Brinkmann, Fast generation of cubic graphs, J. Graph Theory 23 (1996), no. 2, [DS01] Dale Daniel and Stephen E. Shauger, A result on the Erdős-Gyárfás conjecture in planar graphs, Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing(Baton Rouge, LA, 2001), vol. 153, 2001, pp [ES81] R.C.EntringerandP.J.Slater,Onthemaximumnumberofcyclesina graph, Ars Combin. 11(1981), [Gui96] David R. Guichard, The maximum number of cycles in graphs, Proceedings of the Twenty-seventh Southeastern International Conference on Combinatorics, Graph Theory and Computing(Baton Rouge, LA, 1996), vol. 121, 1996, pp [McK] Brendan McKay, The program geng is available from Brendan McKays home page at [Roy] Gordon Royle, The 2ˆn conjecture, gordon/remote/erdosconj.html. [Sha98] StephenE.Shauger,ResultsontheErdős-GyárfásconjectureinK 1,m -free graphs, Proceedings of the Twenty-ninth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1998), vol. 134, 1998, pp [Tut46] W. T. Tutte, On Hamiltonian circuits, J. London Math. Soc. 21(1946), [Wor99] N. C. Wormald, Models of random regular graphs, Surveys in combinatorics, 1999(Canterbury), London Math. Soc. Lecture Note Ser., vol. 267, Cambridge Univ. Press, Cambridge, 1999, pp address: Klas.Markstrom@math.umu.se Department of Mathematics, UmeåUniversity, SE Umeå, Sweden 15