Almost all Cayley Graphs Are Hamiltonian

Transcription

1 Acta Mathmatca Sca, Nw Srs 199, Vol1, No, pp Almost all Cayly Graphs Ar Hamltoa Mg Jxag & Huag Qogxag Abstract It has b cocturd that thr s a hamltoa cycl vry ft coctd Cayly graph I spt of th dffculty provg ths coctur, w show that almost all Cayly graphs ar hamltoa That s, as th ordr of a group G approachs fty, th rato of th umbr of hamltoa Cayly graphs of G to th total umbr of Cayly graphs of G approachs 1 Kywords Cayly graph, Hamltoa, Radom 1 Itroducto Lt G b a group ad S b a vrs-closd subst (S 1 = S ofg ot cotag th dtty lmt 1 of G Th Cayly graph of G wth rspct to S, dotd by Cay(S : G, s a smpl graph whos vrtcs ar th lmts of G, wth a dg og x ad y f x 1 y S It was cocturd by Lovasz [1, p 497] that vry coctd vrtx trastv graph has a hamltoa path Appartly, oly four o-trval coctd vrtx trastv graphs wthout hamltoa cycls ar kow [] All four possss hamltoa paths Howvr, o of ths four graphs s a Cayly graph Ths sprd may popl to coctur that vry coctd Cayly graph wth mor tha two vrtcs has a hamltoa cycl (T D Parsos ad othrs [] W ar owhr ar a proof of ths coctur Ev for trvalt Cayly graphs of dhdral group, ths coctur has ot b sttld [4 5] Usg th wll-kow Jackso s thorm o hamltoa cycls rgular graphs, howvr, w prov th followg Thorm 1 Almost all Cayly graphs ar hamltoa Prlmars All groups ths papr ar ft ad hav at last thr lmts G always dots a ft group Our graph ad group otatos ar trly cosstt wth thos of [] ad [7] I ths scto, w wll stablsh th radom Cayly graph modl ad gv som wll-kow rsults whch wll b usd latr W start wth umratg th lablld Cayly graphs of a gv group Lt G b a group of ordr havg m lmts of ordr Lt V 0 b th st of lmts of ordr Th for ay x G ({1} V 0,x x 1 Partto G ({1} V 0 to two substs V 1 ad V1 such that x V 1 ff x 1 V1 Th V 0 = m ad V 1 = V1 = m 1 Rcvd March 7, 1994 Supportd by th Natoal Natural Scc Foudato of Cha, Xag Educatoal Commtt ad Xag Uvrsty

2 15 Acta Mathmatca Sca, Nw Srs Vol1 No For ay subst S of V 0 V 1, w obta a vrs-closd subst S as follows: S = S {x 1 : x S V 1 } Covrsly, ay vrs-closd subst of G {1} ca b obtad ths way Thus thr ar xactly V0 V1 = +m 1 lablld Cayly graphs of G W thus dduc th followg Lmma If G s a group of ordr ad has m lmts of ordr, th thr ar xactly +m 1 lablld Cayly graphs W ow prst a modl of radom Cayly graphs whch s actually a probablty spac basd o a group Gv a group G of ordr wth m lmts of ordr W us Ω(G to dot th probablty spac cosstg of all lablld Cayly graphs of G, ad ach such graph s assgd th sam probablty +m 1 A radom Cayly graph s cosdrd as a sampl lmt of Ω(G Lt Q b a graph proprty W say that almost all Cayly graphs hav proprty Q f for ay group of ordr, th rato of th umbr of Cayly graphs of G havg th proprty Q to th total umbr of Cayly graphs of G approachs 1 as approachs fty For dtals o radom graphs [8] For otatos about probablty thory rfr to [9], ad for radom Caylay graphs to [10] I provg Thorm 1, w wll us th followg kow rsult: Thorm (Jackso [1] Evry -coctd, k-rgular graph o at most k vrtcs s hamltoa Th followg rsult was frst cocturd by W Holsztysk ad R F E Strub [1] ad was provd by D Wtt [1] Thorm [] Thr s a hamltoa cycl vry coctd Cayly graph of ft p-groups Th umbr of lmts of ordr a group s mportat th followg dscusso Hr w ct a gral rsult: Thorm 4 [14] Lt G b a ft group ad p a prm dvsor of th ordr of G IfG s ot a p p-group, th thr ar at most p +1 G lmts of G satsfyg xp =1 Corollary 5 If G s ot a -group, th thr ar at most G lmts of ordr I our calculatos, w shall us th followg rathr sharp form of Strlg formula provd by Robbss (s [8] p4 or [15] (! = π α, whr 1/(1 +1<α < 1/(1 Proof of th Ma Rsult I ths scto, w frst show that almost all Cayly graphs ar coctd, ad th gv th proof of Thorm 1 Thorm Almost all Cayly graphs ar coctd To prov Thorm, w stablsh a squc of lmmas Th followg Lmma 7 s wll kow, ad s asy to prov Lmma 7 Cayly graph Cay(S : G s coctd ff S grats G

3 Mg Jxag t al Almost All Cayly Graphs Ar Hamltoa 15 By Lmma 7, Cay(S : G s ot coctd ff S s cotad a propr subgroup of G Thus, to stmat th umbr of o-coctd Caylay graphs of G, w d to assss th umbr of vrs-closd substs that ar cotad propr subgroups of G, ad thrfor d to assss th umbr of subgroups of gv ordr To ths d, w prov th followg Lmma 8 Evry group of ordr ca b gratd by at most log lmts Proof Lt G b a group of ordr ad M = {g 1,g,,g k } b a mmal gratg st of G W prov that k log For ay o-mpty subst U of M, mak product g u of th lmts U a ordr such that th dx from lft to rght s crasg Th by th mmalty of M ad th cacllato law of group, w kow that g u g v whvr U V ThusGhas at last k lmts, that s k Ths provs th lmma Lmma 9 Thr ar at most log k subgroups of ordr k ay group of ordr ( Proof Th umbr of subgroups of ordr k s obvously o mor tha log k, ad thrfor o mor tha log k Now w prov Thorm ProofofThorm Lt G b a group of ordr havg m lmts of ordr W stmat th umbr of o-coctd Cayly graphs of G By Lmma 9 ad Lmma, th umbr of vrs-closd substs of G that ar cotad subgroups of ordr k s at most log k k+m 1 Thus th umbr of o-gratg vrs-closd substs of G {1} s o mor tha log k k+m 1 log k+m 1 log +m 4 1k< k 1k< k Lt E dot th vt that a Cayly graph of G s coctd Th th probablty P (E of th complmtary vt E of E s at most Clarly, log lm 4 log +m 4 +m 1 = log 4 =0 Thus lm P (E =0, ad so lm P (E =1, Ths complts th proof Cosdr th Cayly graph Cay(S : G For ay a G, th lft multplcato τ α : x ax(x G s clarly a automorphsm of Cay(S : G, ad all ths lft multplcatos costtut a trastv subgroup of th automorphsm group of Cay(S : G; thus th Cayly graphs ar vrtx trastv O th othr had, coctd vrtx trastv graphs o at last thr vrtcs ar -coctd [1] From Thorm, w dduc th followg Corollary 10 Almost all Cayly graphs ar -coctd Combg Thorm wth a thorm of Watks [1] whch stats that th dg-coctvty of a coctd vrtx trastv graph s ts rgular dgr, w obta Corollary 11 Th dg-coctvty of almost all Cayly graphs s thr rgular dgr Bfor provg Thorm 1, w gv( som smpl qualts o combatos Lmma 1 ( For ay k, 0 k, ( ( ( k ( k 1 k k 1 + k

4 154 Acta Mathmatca Sca, Nw Srs Vol1 No ( ( ( St α = 1 Th for a suffctly larg, α Proof ( s dducd drctly from th dtcal qualty ( = To prov (, k 0k ( 1 + t suffcs to ot that s th umbr of substs wth cardalty k 1 + k of a k 1 + ( k ( 1 ( 1 + -lmt st, whl s that of som of ths substs Now w us th k 1 k Strlg formula to prov ( St l = Th ( =! l!( l! ( 1 1 ( ( π ( l l ( l lπ ( lπ +1 ( 1 ( +1 ( ( l l l ( l l = α Ths complts th proof Now w ar a posto to prov Thorm 1 Proof of Thorm 1 Lt G b a group of ordr havg m lmts of ordr, whr,m 0 By Thorm ad Thorm w kow that almost all Cayly graphs of p- groups ar hamltoa Thus w ca assum th followg that G s ot a -group Thus by Thorm 4, w hav m (1 Rcall that V 0 s th st of lmts of ordr ad V 1 V1 s a partto of G (V 0 {1} such that x V 1 ff x 1 V1 V 0 = m, V 1 = V1 = m 1 Ay vrs-closd subst of G ot cotag 1 ca b ducd by a subst of V 0 V 1 Now for ay S V 0 V 1, th corrspodg vrs-closd subst S has cardalty k = S V 0 + S V 1, ad thrfor th Cayly graph Cay(S : G sk-rgular Thus th umbr of Cayly graphs of dgr k s +=k 0k +=k ( m 1 ( By Thorm, th umbr of coctd o-hamltoa Cayly graphs of G s o mor tha ( m 1 ( By Lmma 1, ( m 1 If m 1, th + = k + If > m 1, th = k ( +m m 1 ( m 1 m +1, ad thus ( m 1 Thus, ths cas, w hav ( +m 1 +m 1 (4 ( m+1 m m+1 (5

5 Mg Jxag t al Almost All Cayly Graphs Ar Hamltoa 155 Th umbr of coctd o-hamltoa Cayly graphs s thrfor o mor tha [( +m 1 ( m+1 ] +m 1 + m m+1 α +m 1 + m α m+1, ( whr α s spcfd Lmma 1 Lt F dot th vt that a Cayly graph of G s hamltoa ad E th vt that a Cayly graph of G s coctd Th P (F (α +m 1 + m α m+1 + P (E +m 1 ( +m 1 = α ( m+1 + α + P (E Not that α < 1, ad from (1 w kow m as, ad from Thorm w hav lm P (E =0 Thus lm P (F =0, ad so lm P (F =1 Ths complts th proof Ackowldgmt Th authors ar gratful to Profssor Zhag Fu- for sprg thm to cosdr th radom Cayly graphs Rfrcs [1] Guy, R, Haa, H, Savr, I V ad Schohm, J ds, Combatoral Structurs ad Thr Applcatos, Gordo ad Brach, Nw York, 1970 [] Body, J A, Hamltoa cycls graphs ad dgraphs, Proc 9th S E Cof Comb, Graph Thory ad Computg, 1978, 8 [] Wtt, D ad Galla, J A, A survy: Hamltoa cycls Cayly graphs, Dscrt Math, 1984, 51: 9 04 [4] Powrs, D L, Excptoal trvalt Cayly graphs for dhdral groups, J Graph Thory, 198, : 4 55 [5] Wag M ad Fag X-gu, A dscrmat codto of H-cycl o dhdral groups, J Sys Sc ad Math Scs, 199, 1: [] Body, J A ad Murty, U S R, Graph Thory wth Applcatos, Nw York, North Hollad, 1979 [7] Hall, M, Th Thory of Groups, Nw York: Macmlla, 1959 [8] Bollobas, B, Radom Graphs, Nw York: Acadmc Prss, 1985 [9] Bllgsly, P, Probablty ad Masur, Nw York: Joh Wly ad Sos, 1979 [10] Baba, L ad Godsl, C D, O th automorphsm groups of almost all Cayly graphs, Europ J Combatorcs, 198, : 9 15 [11] Jackso, B, Hamltoa cycls rgular -coctd graphs, J Comb Thory, Sr B 1980, 9: 7 4 [1] Holsztysk, W ad Strub, R F E, Paths ad crcuts ft groups, Dscrt Math, 1978, : 7 [1] Wtt, D, Cayly dagrams of prm-powr ordr ar hamltoa, J Comb Thory, 198, 408: [14] Laffy, T T, Th umbr of soluto of x p = 1 a ft group, Math Proc Combrdg Phlos Soc, 197, 80: 9 1 [15] Robbs, H, A rmark o strlg s formula, Amr Math Mothly, 1955, : 9 [1] Watks, M E, Coctvty of trastv graphs, J Comb Thory, 1970, 8: 9 Mg Jxag & Huag Qogxag Dpartmt of Mathmatcs ad Isttut of Mathmatcs ad Phscs Xag Uvrsty Urumq, 8004 Cha