Square of Hamilton cycle in a random graph

Transcription

1 Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs th prvious rsults of Kühn and Osthus and also Nnadov and Škorić. 1 Introduction In this not w only considr binomial random graphs. Th binomial random graph G n,p is th random graph G with vrtx st [n] in which vry pair {i, j} ( ) [n] 2 appars indpndntly as an dg in G with probability p. W say that a squnc of vnts E n in a probability spac holds with high probability (or w.h.p.) if th probability that E n holds tnds to 1 as n. Throughout this not all logarithms ar natural (bas ) and all asymptotics ar takn in n. By th kth powr of a Hamilton cycl, w man a prmutation v 1, v 2,..., v n of [n] such that {v i, v j } E(G) whnvr i < j i + k. (Hr i + k is to b takn as i + k n if i + k n + 1.) Hamilton cycls hav long bn studid in th contxt of random graphs (s,.g., [1, 2, 5, 9]). Powrs of Hamilton cycls ar lss wll-studid and much lss is known about thm. Sinc th kth powr of a Hamilton cycl contains kn dgs, w can s that if Y k dnots th numbr of copis of such, thn by using Stirling s formula E(Y k ) = 1 2 (n ( n ) ( ) n np 1)!pkn 1 2 2πn p kn = 1 k n 2 2πn (1) ( (1 ε) n and so E(Y k ) 0 if p w.h.p. G n,p contains no kth powr of a Hamilton cycl. ) 1/k ( ) for any constant ε > 0. Thus, if p (1 ε) 1/k, n thn Dpartmnt of Mathmatics, Dpartmnt of Mathmatics, Wstrn Michigan Univrsity, Kalamazoo, MI. Supportd in part by Simons Foundation Grant # and by th National Scurity Agncy undr Grant Numbr H Th Unitd Stats Govrnmnt is authorizd to rproduc and distribut rprints notwithstanding any copyright notation hron. Dpartmnt of Mathmatical Scincs, Carngi Mllon Univrsity, Pittsburgh PA. Rsarch supportd in part by NSF grant CCF and Simons Foundation Grant #

2 Kühn and Osthus [6] obsrvd that for k 3, p ( 1/k n) is th corrct thrshold for th xistnc of th kth powr of a Hamilton cycl. This coms dirctly from a rsult of Riordan [10]. For k = 2 thy gav a bound of p n 1/2+ε (for any ε > 0) bing sufficint for th xistnc of th squar of a Hamilton cycl w.h.p.. This rsult was improvd by Nnadov and Škorić [8] to p C log4 n n (C is a positiv constant) bing sufficint for th xistnc of th squar of a Hamilton cycl. Hr w prov a tight rsult. Thorm 1 Suppos that np 2 =, whr > 0 is a constant. (i) If < thn w.h.p. G n,p dos not contain th squar of a Hamilton cycl. (ii) If > thn w.h.p. G n,p contains th squar of a Hamilton cycl. Th proof is basd on a dlicat application of th scond momnt mthod. 2 Proof Part (i) immdiatly follows from (1). Th rmaining part (ii) will follow from th scond momnt mthod. W say that a prmutation π of [n] in G n,p is squar inducing if G n,p contains dgs {i, π(i)} and { i, π 2 (i) } for ach i [n]. For i [n], lt d(i, π) b th numbr of vrtics j such that ithr (a) {i, π(i), j} or (b) {i, π 2 (i), j} forms a triangl. St γ = log n. W say that a squar inducing prmutation π is good if d(i, π) γ for all i [n]. Lt X b th random variabl that counts th numbr of good prmutations. Obsrv nxt that th Chrnoff bounds imply that Pr(d(i, π) γ π) 2Pr(Bin(n, p 2 ) γ) 2 (for th inquality s,.g., Thorm 21.9 in [4]). ( ) γ = o(n 1 ), γ First w show that. Th FKG inquality (s,.g., Thorm 21.5 in [4]) implis that n!p 2n (1 o(n 1 )) n n!p 2n ( ) np 2 n 2πn = ( ) n 2πn, which clarly gos to infinity. 2

3 Now w hav a choic. W can condition on all squar inducing prmutations bing good or just counting good prmutations. Th computations ar th sam and w hav plumpd for th lattr. In th rnaming part of th proof w show that Pr(X = 0) = o(1) by using th Chbyshv inquality. Fix a good prmutation π. Lt H(π) = (1, π(1), π 2 (1),..., π n 1 (1), 1) b th Hamilton cycl inducd by π. (Th dg st of H(π) is E(H) = {{i, π(i)} : i [n]}.) Thn lt N(a, b, c) b th numbr of good prmutations π such that: (a) E(H(π)) E(H( π)) = b, (b) E(H(π)) E(H( π)) consists of a vrtx-disjoint paths, say P 1, P 2,..., P a, and (c) thr ar xactly c dgs of th form {i, π 2 (i)} in th squar of a Hamilton cycl inducd by π which ar not in th squar of P j for any j [a]. Obsrv that N(a, b, c) dos not dpnd on π and 0 c n (b a). Not that E(X 2 ) n!n(0, 0, 0)p4n Sinc trivially, N(0, 0, 0) n!, w obtain, E(X 2 ) n n b b c=0 c=0 n!n(a, b, c)p 4n (2b+c a) 2. N(a, b, c)p 2n (2b+c a). (2) W will show that th lattr is o(1). Consquntly, th Chbyshv inquality implis that as rquird. Pr(X = 0) E(X2 ) 2 1 = o(1), It rmains to show that th tripl summation in (2) is o(1). First w find an uppr bound on N(a, b, 0) and thn w mak corrctions for th cas c > 0. Choos a vrtics v i, 1 i a, on π. W hav at most choics. Lt n a b 1 + b b a = b, whr b i 1 is an intgr for vry 1 i a. Not that this quation has xactly ( ) b 1 3

4 solutions. For vry i, w choos a path of lngth b i in H(π) which starts at v i. Thus, by th abov considration w can find a vrtx-disjoint paths in H(π) with th total of b dgs in at most ( ) b 1 n a (3) many ways. Lt P 1, P 2,..., P a b any collction of th abov a paths. Now w count th numbr of prmutations π containing ths paths. W s ach dg of P i in at most 2 ordrs. Crudly, vry such squnc can b chosn in at most 2 a ways. Now w bound th numbr of prmutations containing ths squncs. First not that Thus w hav n V (P i ) = b i + 1. a (b i + 1) = n b a i=1 vrtics not in V (P 1 ) V (P a ). W choos a prmutation σ of V \ (V (P 1 ) V (P a )). Hr w hav at most (n b a)! choics. Consquntly, th numbr of prmutations containing P 1, P 2,..., P a is smallr than (2γ) a (n b a)!. (4) Th factor γ a bounds th numbr of choics for π(j), whn j is th nd of a path. Thus, by (3) and (4) and th Stirling formula w obtain ( ) ( ) b 1 b 1 2πn ( n ) n b a N(a, b, 0) (2γn) a (n b a)! (2γn) a. Sinc w gt Hnc, n!p 2n ( n ) n 2πn p 2n, N(a, b, 0)p 2n (2b a) ( ) b 1 ( ) b+a ( ) ( ) b b 1 (2γn) a p a 2b = (2γp) a. n n b N(a, b, 0)p 2n (2b a) n 2γp b ( ) ( ) b b 1 (2γp) a ( ) b (1 + 2γp) = o(1), b=1 4

5 sinc b ( b 1 a=1 a 1) (2γp) a 1 = (1 + 2γp) b 1. W now dal with th cas c > 0. W can account for this by rplacing (4) by ( ) n b a (2γ) a γ c (n b a 2c)!. c This is bcaus ach i contributing to th count c will rduc th numbr of choics for π(i) ithr whn (a) π 2 (i)) = π(i) or whn (b) π(i) = π 2 (i) in th altrnativ cas. W hav coordinatd (a), (b) hr with (a), (b) in th dfinition of d(i, π). In cas (a) j = π(i) and in cas (b) j = π(i). Not that it is not possibl for i to contribut to both cass. Thr sms to b th possibility that thr can b ovrlap in that w can hav π(i) = π 2 (i) and π(i) = π 2 (i) but this cas will corrspond to a path of lngth on whr π, π travrs th dg {π(i), π 2 (i)} in opposit dirctions. Thus this will not contribut to th count c, but instad to a and b. Th binomial cofficint coms from choosing th st of indics i. It follows from th abov that Consquntly, N(a, b, c) N(a, b, 0) (2γ)a γ c (n b a 2c)!. c!(n b a c)! n b N(a, b, c)p 2n (2b+c a) n b ( ) ( b b 1 )(2γp) a γ c (n b a 2c)! c!p c (n b a c)! (5) Now if c n 2/3, thn c! (c/) c and so c=n 2/3 If c n 2/3 and n b a n 3/4, thn n 2/3 γ c (n b a 2c)! c!p c (n b a c)! c=n 2/3 ( ) γ c = o(1). cp γ c ( ) (n b a 2c)! 2γ c c!p c (n b a c)! n 3/4 = o(1). p Finally, if c n 2/3 and n b a n 3/4, which implis that b (n n 3/4 )/2 sinc b a, thn w hav n b=(n n 3/4 )/2 a=1 b ( ) ( n b b 1 )(2γp) 2/3 a γ c c!p c n2 ( ) ( ) (n n 3/4 )/2 γ n 2/3 = o(1). p This provs that th R.H.S. of (5) is o(1) and complts th proof of part (ii) of Thorm 1. 5

6 3 Final Rmarks Using th argumnt of McDiarmid [7] w obtain th sam rsult for dirctd graphs. It follows from th mbdding thorm in Dudk, Friz, Ruciński and Šilikis [3] that w.h.p. th random r-rgular G n,r contains th squar of a Hamilton cycl as long as n log n r n. Rfrncs [1] M. Ajtai, J. Komlós and E. Szmrédi, Th first occurrnc of Hamilton cycls in random graphs, Annals of Discrt Mathmatics 27 (1985), [2] B. Bollobás, Th volution of spars graphs, in Graph Thory and Combinatorics, Acadmic Prss, Procdings of Cambridg Combinatorics, Confrnc in Honour of Paul Erdős (B. Bollobás; Ed) (1984), [3] A. Dudk, A.M. Friz, A. Ruciński and M. Šilikis, Embdding th Erdős-Rényi hyprgraph into th random rgular hyprgraph and Hamiltonicity, arxiv: v1. [4] A.M. Friz and M. Karoński, Introduction to random graphs, Cambridg Univrsity Prss, [5] J. Komlós and E. Szmrédi, Limit distributions for th xistnc of Hamilton circuits in a random graph, Discrt Mathmatics 43 (1983), [6] D. Kühn and D. Osthus, On Pósa s conjctur for random graphs, SIAM Journal on Discrt Mathmatics 26 (2012), [7] C. McDiarmid, Cluttr prcolation and random graphs, Mathmatical Programming Studis 13 (1980), [8] R. Nnadov and N. Škorić, Powrs of cycls in random graphs and hyprgraphs, arxiv: v1. [9] L. Pósa, Hamiltonian circuits in random graphs, Discrt Mathmatics 14 (1976), [10] O. Riordan, Spanning subgraphs of random graphs, Combinatorics, Probability and Computing 9 (2000),