Packing, Counting and Covering Hamilton cycles in random directed graphs

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1 Packig, Coutig ad Coverig Hamilto cycles i radom directed graphs Asaf Ferber Gal Kroeberg Eoi Log November 12, 2015 Abstract A Hamilto cycle i a digraph is a cycle that passes through all the vertices, where all the arcs are orieted i the same directio. The problem of fidig Hamilto cycles i directed graphs is well studied ad is kow to be hard. Oe of the mai reasos for this, is that there is o geeral tool for fidig Hamilto cycles i directed graphs comparable to the so called Posá rotatio-extesio techique for the udirected aalogue. Let D(, p) deote the radom digraph o vertex set [], obtaied by addig each directed edge idepedetly with probability p. Here we preset a geeral ad a very simple method, usig kow results, to attack problems of packig ad coutig Hamilto cycles i radom directed graphs, for every edge-probability p > log C ()/. Our results are asymptotically optimal with respect to all parameters ad apply equally well to the udirected case. 1 Itroductio A Hamilto cycle i a graph or a directed graph is a cycle passig through every vertex of the graph exactly oce, ad a graph is Hamiltoia if it cotais a Hamilto cycle. Hamiltoicity is oe of the most cetral otios i graph theory, ad has bee itesively studied by umerous researchers i the last couple of decades. The decisio problem of whether a give graph cotais a Hamilto cycle is kow to be N P-hard ad is oe of Karp s list of 21 N P-hard problems [23]. Therefore, it is importat to fid geeral sufficiet coditios for Hamiltoicity ad ideed, may iterestig results were obtaied i this directio. Oce Hamiltoicity has bee established for a graph there are may questios of further iterest. For example, the followig are atural questios: Let G be a graph with miimum degree δ(g). Is it possible to fid roughly δ(g)/2 edge-disjoit Hamilto cycles? (This problem is referred to as the packig problem.) Departmet of Mathematics, Yale Uiversity, ad Departmet of Mathematics, MIT. s: asaf.ferber@yale.edu, ad ferbera@mit.edu. School of Mathematical Scieces, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv, , Israel. galkroe@mail.tau.ac.il. School of Mathematical Scieces, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv, , Israel. eoilog@post.tau.ac.il. 1

2 Let (G) deote the maximum degree of G. Is it possible to fid roughly (G)/2 Hamilto cycles for which every edge e E(G) appears i at least oe of these cycles? (This problem is referred to as the coverig problem.) How may distict Hamilto cycles does a give graph have? (This problem is referred to as the coutig problem.) All of the above questios have a log history ad may results are kow. Let us defie G(, p) to be the probability space of graphs o a vertex set [] := {1,..., }, such that each possible (uordered) pair xy of elemets of [] appears as a edge idepedetly with probability p. We say that a graph G G(, p) satisfies a property P of graphs with high probability (w.h.p.) if the probability that G satisfies P teds to 1 as teds to ifiity. Packig. The questio of packig i the probabilistic settig was firstly discussed by Bollobás ad Frieze i the 80 s. They showed i [4] that if {G i } ( 2) i=0 is a radom graph process o [], where G 0 is the empty graph ad G i is obtaied from G i 1 by adjoiig a o-edge of G i 1 uiformly at radom, as soo as G i has miimum degree k (where k is a fixed iteger), it has k/2 edge-disjoit Hamilto cycles plus a disjoit perfect matchig if k is odd. This result geeralizes a earlier result of Bollobás [3] who proved (amog other thigs) that for p =, a typical graph G G(, p) is l +l l +ω(1) l +l l ω(1) Hamiltoia. Note that this value of p is optimal i the sese that for p =, it is kow that w.h.p. a graph G G(, p) satisfies δ(g) 1, ad therefore is ot Hamiltoia. Later o, Frieze ad Krivelevich showed i [14] that for p = (1 + o(1)) l, a graph G G(, p) w.h.p. cotais δ(g)/2 edge-disjoit Hamilto cycles (i fact, this was prove oly usig pseudo-radom hypothesis), which has afterwards bee improved by Be-Shimo, Krivelevich ad Sudakov i [2] to p 1.02 l. We remark that i this regime of p, w.h.p. G G(, p) is quite far from beig regular. As the culmiatio of a log lie of research Kox, Küh ad Osthus [24], Krivelevich ad Samotij [26] ad Küh ad Osthus [28] completely solved this questio for the etire rage of p. For the o-radom case, it is worth metioig a recet remarkable result due to Csaba, Küh, Lo, Osthus ad Treglow [5] which proved that for large eough ad d /2, every d-regular graph o vertices cotais d/2 edge-disjoit Hamilto cycles ad oe disjoit perfect matchig i case d is odd. This result settles a log stadig problem due to Nash-Williams [31] for large graphs. Coverig. The problem of coverig the edges of a radom graph was firstly studied i [18] by Glebov, Krivelevich ad Szabó. It is show that for p 1+ε, the edges of a typical G G(, p) ca be covered by (1 + o(1))p/2 edge-disjoit Hamilto cycles. Furthermore they proved aalogous results also i the pseudo-radom settig. I [19], Hefetz, Lapiskas, Küh ad Osthus improved it by showig that for some C > 0 ad logc () p 1 1/8, oe ca cover all the edges of a typical graph G G(, p) with (G)/2 Hamilto cycles. Coutig. Give a graph G, let h(g) deote the umber of distict Hamilto cycles i G. Stregtheig the classical theorem of Dirac from the 50 s [8], Sárközy, Selkow ad Szemerédi [33] proved that every graph G o vertices with miimum degree at least /2 cotais ot oly oe but at least c! Hamilto cycles for some small positive costat c. They also cojectured that this c could be improved to 1/2 o(1). This was later prove by Cuckler ad Kah [7]. I fact, Cuckler ad Kah proved a stroger result: every graph G o vertices with miimum degree δ(g) /2 2

3 ( ) has h(g) δ(g) e (1 o(1)). A typical radom graph G G(, p) with p > 1/2 shows that this estimate is sharp (up to the (1 o(1)) factor). Ideed, i this case with high probability δ(g) = p + o() ad the expected umber of Hamilto cycles is p ( 1)! < (p/e). I the radom/pseudo-radom settig, buildig o ideas of Krivelevich [25], i [17] Glebov ad l +l l +ω(1) Krivelevich showed that for p ad for a typical G G(, p) we have h(g) = (1 o(1))!p. That is, the umber of Hamilto cycles is, up to a sub-expoetial factor, cocetrated aroud its mea. For larger values of p, Jaso showed [21] that the distributio of h(g) is log-ormal, for G G(, p) with p = ω( 1/2 ). I this paper we treat the three of these problems i the radom directed settig. A directed graph (or digraph) is a pair D = (V, E) with a set of vertices V ad a set of arcs E, where each arc is a ordered pair of elemets of V. A directed graph is called orieted, if for every pair of vertices u, v V, at most oe of the directed edges uv or vu appears i the graph. A touramet is a orieted complete graph. A Hamilto cycle i a digraph is a cycle goig through all the vertices exactly oce, where all the arcs are orieted i the same directio i a cyclic order. Give a directed graph D ad a vertex v V, we let d + D (v) ad d D (v) deote its out- ad i- degree i D. Let D(, p) be the probability space cosistig of all directed graphs o vertex set [] i which each possible arc is added with probability p idepedetly at radom. The problem of determiig the rage of values of p for which a typical graph D D(, p) is Hamiltoia goes back to the early 80 s, where McDiarmid [30] showed, amog other thigs, that a elegat couplig argumet gives the iequality Pr[G G(, p) is Hamiltoia] Pr[D D(, p) is Hamiltoia]. Combied with the result of Bollobás [3] it follows that a typical D D(, p) is Hamiltoia for l +ω(1) p. Later o, Frieze showed i [16] that the same coclusio holds for p l +l l +ω(1) l ω(1). The result of Frieze is optimal i the sese that for p =, it is ot difficult to see that for a typical D D(, p) we have mi v V {δ + (v), δ (v)} = 0 ad therefore D is ot Hamiltoia. Robustess of Hamilto cycles i radom digraphs was studied by Hefetz, Steger ad Sudakov i [20] ad by Ferber, Neadov, Noever, Peter ad Skorić i [13]. 1.1 Our results While i geeral/radom/pseudo-radom graphs there are may kow results, much less is kow about the problems of coutig, packig ad coverig i the directed settig. The mai difficulty is that i this settig the so called Posá rotatio-extesio techique (see [32]) does ot work i its simplest form. I this paper we preset a simple method to attack ad approximately solve all the above metioed problems i radom/pseudo-radom directed graphs, with a optimal (up to a polylog() factor) desity. Our method is also applicable i the udirected settig, ad therefore reproves may of the above metioed results i a simpler way. The problem of packig Hamilto cycles i digraphs goes back to the 70 s. Tilso [36] showed that every complete digraph has a Hamilto decompositio. Recetly, a remarkable result of Küh ad Osthus (see [27]) proves that for ay regular orietatio of a sufficietly dese graph oe ca fid a Hamilto decompositio. I the case of a radom directed graph, ot much is kow regardig 3

4 packig Hamilto cycles. Our first result proves the existece of (1 o(1))p edge-disjoit Hamilto cycles i D(, p). ( ) Theorem 1.1. For p = ω log 4, w.h.p. the digraph D D(, p) has (1 o(1))p edge-disjoit Hamilto cycles. We also show that i radom directed graphs oe ca cover all the edges by ot too may cycles. ) Theorem 1.2. Let p = ω. The, a digraph D D(, p) w.h.p. ca be covered with (1 + ( log 2 o(1))p directed Hamilto cycles. The problem of coutig Hamilto cycles i digraphs was already studied i the early 70 s by Wright i [38]. However, coutig Hamilto cycles i touramets is a eve older problem which goes back to oe of the first applicatios of the probabilistic method by Szele [34]. He proved that there are touramets o vertices with at least ( 1)!/2 Hamilto cycles. Thomasse [35] cojectured that i fact every regular touramet cotais at least (1 o(1)) Hamilto cycles. This cojecture was solved by Cuckler [6] who proved that every regular touramet o vertices! cotais at least (2+o(1)) Hamilto cycles. Ferber, Krivelevich ad Sudakov [11] later exteded Cuckler s result for every early c-regular orieted graph for c > 3/8. Here, we cout the umber of Hamilto cycles i radom directed graphs ad improve a result of Frieze ad Sue from [15]. We show that the umber of directed Hamilto cycles i such radom graphs is cocetrated (up to a sub-expoetial factor) aroud its mea. Theorem 1.3. Let p = ω directed Hamilto cycles. ( log 2 ). The, a digraph D D(, p) w.h.p. cotais (1 ± o(1))!p Fially, the same proof method ca be used to prove aalogous results whe istead workig with pseudo-radom graphs. We direct the reader to Defiitio 6.1 i Sectio 6.1 for the otio of pseudo-radomess used here. The followig theorems show that at a cost of a additioal polylog factor i the desity we obtai aalogues of Theorem 1.1, 1.2, 1.3 for pseudo-radom digraphs. Below we will write o λ (1) for some quatity tedig to 0 as λ 0. ( ) Theorem 1.4. Let D be a (, λ, p) pseudo-radom digraph where p = ω log 14. The D cotais (1 o λ (1))p edge-disjoit Hamilto cycles. Theorem 1.5. Let D be a (, λ, p) pseudo-radom digraph where p = ω covered with (1 + o λ (1))p directed Hamilto cycles. Theorem 1.6. Let D be a (, λ, p) pseudo-radom digraph where p = ω cotais (1 o λ (1))!p directed Hamilto cycles. ( ) log 14. The D ca be ( ) log 14. The D ca be We have oly icluded the proof of Theorem 1.4 which modifies the proof of Theorem 1.1 to the pseudo-radom settig. The other results ca be prove i a similar maer (these other proofs are i fact slightly easier). Remark 1.7. We also draw attetio to the fact that all of our proofs also apply to G(, p) with the same probability thresholds as i Theorem 1.1, 1.2 ad 1.3. Although all these results are kow i G(, p) (ad i fact eve much more), our approach provides us with short ad elegat proofs. For coveiece, we state the exact statemets which follow from our proofs: 4

5 For p = ω ( log 4 ) our approach gives that G G(, p) whp cotais (1 o(1))p/2 edge disjoit Hamitlo cycles. As metioed i the packig sectio above, here it is kow that for all p whp G G(, p) cotais δ(g)/2 edge disjoit Hamilto cycles (see [24], [26] ad [28]). For p = ω ( log 2 ) our approach gives that G G(, p) whp cotais (1 + o(1))p/2 Hamilto cycles coverig all edges of G. As metioed i the coverig sectio above, here it is kow that there is some costat C > 0 such that for logc p 1 1/18 whp G G(, p) has a edge coverig with (G)/2 Hamilto cycles (see [19]). For p = ω ( log 2 ) our approach gives that G G(, p) whp cotais (1 ± o(1))!p Hamilto cycles. As metioed i the coutig sectio above, here it is kow that such a boud already applies for p > log +log log +ω(1). 1.2 Notatio ad termiology We deote by D the complete directed graph o vertices (that is, all the possible ( 1) arcs appear), ad by D,m the complete bipartite digraph with parts [] ad [m]. Give a directed graph F ad a vector p (0, 1] E(F ), we let D(F, p) deote the probability space of sub-digraphs D of F, where for each arc e E(F ), we add e ito E(D) with probability p e, idepedetly at radom. I the special case where p e = p for all e, we simply deote it by D(F, p). I the case where F = D, we write D(, p) ad i the case F = D,m we write D(, m, p). Give a digraph D ad two sets X, Y V (D) we write E D (X, Y ) = { xy E(D) : x X, y Y }. Also let e D (X, Y ) = E D (X, Y ) ad e D (X) = E D (X, X). We will also occasioally make use of the same otatio for graphs G, i.e. e G (X, Y ). For a vertex v we deote N + D (v) = E D({v}, V (D)) ad N D (v) = E D(V (D), {v}). Let d + D (v) = N + D (v) ad d D (v) = N D (v). Lastly, we write x a ± b to mea that x is i the iterval [a b, a + b]. 2 Overview ad auxiliary results 2.1 Proof overview Our aim i this subsectio is to provide a overview of the proofs of Theorems 1.1, 1.2 ad 1.3. I particular, we hope to highlight the similarities ad differeces which occur for the packig, coutig ad coverig problems. To do this, we will first describe a approach to solve similar problems for a more restricted model of radom digraph. We the outlie how these results ca be used to solve the correspodig problems for D(, p). Suppose that we are give a partitio [] = V 0 V 1 V l, with V 0 = s ad V j = m for all j [l] so that = ml + s (here s = ω(m) ad l = polylog()). Cosider the followig way to select radom digraph F : 1. For all j [l 1], directed edges from V j ad V j+1 are adjoied to F with probability p i idepedetly. Let F j deote this sub-digraph of F ; 2. The directed edges (a) i V 0 (b) from V 0 to V 1 (c) from V l to V 0 ad (d) from V l to V 1 are adjoied to F with probability p ex idepedetly. Let F 0 deote this subdigraph of F. 5

6 This selectio process gives a distributio o a set of digraphs. We will write F to deote this distributio, ad write F F to deote a digraph F chose accordig to it. We will describe how to show that if F F the whp, for appropriate values of p i ad p ex, we have: (i)* (1 o(1))mp i edge disjoit Hamilto cycles which cotai almost all edges of type 1 i F ; (ii)* (1 + o(1))mp i Hamilto cycles which cover all edges of type 1 i F ; (iii)* (1 o(1)) s (m!) l 1 p s m i directed Hamilto cycles i F. To do this, we first expose edges of type 1. above. Usig kow matchig results, for p i = ω(log C m/m) ad l m say, it ca be show that whp for every j [l 1]: (i) F j cotais L pack := (1 o(1))mp i edge disjoit perfect matchigs, {M j i }L pack i=1 ; (ii) F j cotais L cov := (1 + o(1))mp i perfect matchigs coverig all edges of F j, {M j i }Lcov i=1 ; (iii) F j cotais (1 o(1)) m m!p m i perfect matchigs. Now ote that i (i), (ii) ad (iii) above, by combiig a perfect matchig from each F j for each j [l 1] we obtai a collectio of m vertex disjoit directed paths from V 1 to V l, coverig j [l] V j. We refer to such a collectio of paths P as a matchig path system. (i) For each i [L pack ], by combiig the disjoit matchigs {M j i }l 1 j=1 from (i) i this way, we obtai a matchig path system P i. This gives L pack edge disjoit matchig path systems P 1,..., P Lpack. (ii) For each i [L cov ], by combiig the matchigs {M j i }l 1 j=1 from (ii) i this way we obtai a matchig path system P i. This gives L cov matchig path systems P 1,..., P Lcov, which cover all edges i the digraphs F j for j [l 1]. (iii) Lastly, by choosig differet matchig betwee the partitios from (iii), we have may choices for how to build our matchig path system P. We obtai at least (1 o(1)) ml (m!) l 1 p m(l 1) i (1 o(1)) (m!) l 1 p s m i such choices for P. Now let P = {P 1,..., P m } be a fixed matchig path system. Assume that each P i begis at a vertex s i V 1 ad termiates at a vertex t i V l. These vertices are distict by costructio. We will ow describe how to iclude all paths i P ito a directed Hamilto cycle. To do this simply cotract each directed path P i to sigle vertex which we also deote by P i. Now expose the edges of type 2. above ad view them as edges of a radom digraph o vertex set Ṽ = V 0 {P 1,..., P m }. Note that the followig edges all appear with probability p ex : All directed edges i V 0. These come from edges of type 2. (a) above; Directed edges from V 0 to {P 1,..., P m } ad from {P 1,..., P m } to V 0. These come respectively from edges of type 2. (b) ad (c) above; Directed edges withi the set {P 1,..., P m }. These come from edges of type 2 (d) above. (Here we may obtai a loop o the vertices P i, which we simply igore.) 6

7 As all such edges appear idepedetly, the resultig radom digraph is distributed idetically to D(s+m, p ex ). By kow Hamiltoicity result for D(, p), provided that p ex = ω ( log C (m + s)/(m + s) ) we obtai that this digraph is Hamiltoia with very high probability. However, it is easy to see that by costructio a directed Hamilto cycle i this cotracted digraph pulls back to a directed Hamilto cycle i F, which cotais the paths i P as directed subpaths. Thus we have show how to tur a sigle matchig path system ito a Hamilto cycle. Now i the case of (ii)*, we ca complete each of the matchig path systems P 1,... P Lcov ito Hamilto cycles by usig edges of type 2. described above. This ca also be used to show whp may of the matchig paths systems from (iii)* complete to (distict) directed Hamilto cycles. However, to pack the Hamilto cycles i the case of (i)* more care must be take as we caot use the same edges twice. To get aroud this, we distribute the edges of type 2. to create a idividual radom digraph for each P i. Provided that p ex is sufficietly large (ad m, l ad s are carefully chose) each of these idividual radom digraphs will be Hamiltoia whp. This completes the descriptio of (i)*, (ii)* ad (iii)* above. Now our approach for dealig with the packig, coverig ad coutig problems o D(, p) is to show that with high probability we ca break D D(, p) ito subdigraphs distributed similarly to F above. However the type of decompositio chose is agai depedet o the problem at had. With the packig it is importat that these graphs are edge disjoit. With the coverig, it will be importat every edge of D(, p) appears as a edge of type 1. i oe of these digraphs (recall these were the oly edges guarateed to be covered i (ii)*). The coutig argumet is less sesitive, ad simply work with may such digraphs. Depedet o the problem, we ca apply our strategy above for F F to each of these digraphs separately. Combiig the resultig Hamilto cycles from either (i)*, (ii)* or (iii)* i each of these digraphs will the solve the correspodig problem for D(, p). 2.2 Probabilistic tools We will eed to employ bouds o large deviatios of radom variables. We will mostly use the followig well-kow boud o the lower ad the upper tails of the biomial distributio due to Cheroff (see [1], [22]). Lemma 2.1 (Cheroff s iequality). Let X Bi(, p) ad let µ = E(X). The Pr[X < (1 a)µ] < e a2 µ/2 for every a > 0; Pr[X > (1 + a)µ] < e a2 µ/3 for every 0 < a < 3/2. Remark 2.2. The coclusios of Cheroff s iequality remai the same whe X has the hypergeometric distributio (see [22], Theorem 2.10). We will also fid the followig boud useful. Lemma 2.3. Let X Bi(, p). The Pr [X k] ( ep k ) k. Proof. Just ote that Pr [X k] ( ) p k k ( ep ) k. k 7

8 2.3 Perfect matchigs i bipartite graphs ad radom bipartite graphs The followig lower boud o the umber of perfect matchigs i a r-regular bipartite graph is also kow as the Va der Waerde cojecture ad has bee prove by Egorychev [9] ad by Falikma [10]: Theorem 2.4. Let G = (A B, E) be a r-regular bipartite graph with parts of sizes A = B =. The, the umber of perfect matchigs i G is at least ( r )!. The followig lemma is a easy corollary of the so called Gale-Ryser theorem (see, e.g. [29]). Lemma 2.5. (Lemma 2.4, [12]) Let G is a radom bipartite graph betwee two vertex sets both of size, where edges are chose idepedetly with probability p = ω(log /). The with probability 1 o(1/) the graph G cotais (1 o(1))p edge disjoit perfect matchigs. 2.4 Covertig paths ito Hamilto cycles The followig defiitios will be coveiet i our proofs. Defiitio 2.6. Suppose that X is a set of size ad that l, m, s are positive itegers with = ml+s. A sequece V = (V 0, V 1,..., V l ) of subsets of X is called a (l, s)-partitio of X if X = V 0 V 1... V l is a partitio of X, ad V 0 = s, ad V i = m for every i [l]. Defiitio 2.7. Give a (l, s)-partitio V = (V 0, V 1,..., V l ) of a set X, let D (V) deote the digraph o vertex set X = [] cosistig of all edges uv such that: 1. u V j, v V j+1 for some j [l 1], or 2. u V 0 ad v V 0 V 1, or 3. u V l, v V 0 V 1. We call edges of type 1. iterior edges ad call edges of type 2. ad 3. exterior edges. Suppose that we are give two disjoit sets V ad W ad a digraph D o vertex set V W. Suppose also that we have m disjoit ordered pairs M = {(w i, x i ) : i [m]} W W. The we defie the followig auxiliary graph. Defiitio 2.8. Let D(M, V ) deote the followig auxiliary digraph o vertex set M V where M = {u 1,..., u m } ad each u i refers to the pair (w i, x i ). The give ay two vertices v 1, v 2 V, we have: v 1 v 2 is a edge i D(M, V ) if it appears i D; v 1 u i is a edge i D(M, V ) if v 1 w i is a edge i D; u i v 1 is a edge i D(M, V ) if x i v 1 is a edge i D; u i u j is a edge i D(M, V ) if x i w j is a edge i D. Remark 2.9. Note that if D(M, V ) cotais a directed Hamilto cycle ad W ca be decomposed ito vertex disjoit directed w i x i -paths for all i [m] (paths startig at w i ad edig at x i ) the D cotais a directed Hamilto cycle. 8

9 3 Coutig Hamilto cycles i D(, p) I this sectio we prove Theorem 1.3. The proof of this theorem is relatively simple ad cotais most of the ideas for the other mai results ad therefore serves as a ice warmup. Proof. We will first prove the upper boud. For this, let X H deote the radom variable that couts the umber of Hamilto cycles i D D(, p). It is clear that E[X H ] = ( 1)!p. By Markov s iequality, we therefore have P r(x H (1 + o(1))!p ) E[X H ] (1 + o(1))!p = (1 o(1)) = o(1). Thus X H (1 + o(1))!p w.h.p.. We ow prove the lower boud, i.e. X H (1 o(1))!p w.h.p.. Let α := α() be a fuctio tedig to ifiity arbitrarily slowly with. We prove the lower boud o X H uder the assumptio that p α 2 log 2 /. Let us take s ad l to be itegers where s is roughly α log ad l is roughly 2α log ad there is a iteger m with = lm + s. Also fix a set S V (G) of order s ad let us set V = V (D) \ S. The set S will be used to tur collectios of vertex disjoit paths ito Hamilto cycles. To begi, take a fixed (l, s)-partitio V = (V 0, V 1,..., V l ) with V 0 = S. We claim the followig: Claim: Give V as above, takig D D(, p), the radom digraph D D (V) (where D (V) is as i Defiitio 2.7) cotais at least (1 o(1)) m! l 1 p m(l 1) distict Hamilto cycles with probability 1 o(1). To see this, first expose the iterior edges of D D (V). For each j [l 1] let F j := E D (V j, V j+1 ). Observe that F j D(m, m, p). It will be coveiet ) for us to view F j as a bipartite graph obtaied by igorig the edge directios. Sice p = ω, by Lemma 2.5 with probability 1 o(1/) we ( log m coclude that F j cotais (1 o(1))mp edge-disjoit perfect matchigs. Takig a uio boud over all j [l 1] we fid that whp F j cotais a (1 o(1))mp-regular subgraph for all j [l 1]. Apply Theorem 2.4 to each of these subgraphs. This give that for each j [l 1] the graph F j cotais at least (1 o(1)) m m!p m perfect matchigs. Combiig a perfect matchig from each of the F j s we obtai a family P of m vertex disjoit paths which spas V. Let Λ V deote the set of all such P. From the choices of perfect matchigs i each F j we obtai that whp Now let P = {P 1,..., P m } Λ V. Let Λ V ((1 o(1)) m m!p m ) l 1 = (1 o(1)) (m!) l 1 p s. (1) M = {(u i, v i ) V 1 V l : P i is a u i v i directed path}. Let us cosider the auxiliary digraph D(M, V 0 ) as i Defiitio 2.8. As we expose the exterior edges of D i D (V) it is easy to see that D(M, V 0 ) D(s + m, p). Furthermore, a Hamilto cycle i D(M, V 0 ) gives a Hamilto cycle i D by Remark 2.9. However, it is well-kow digraphs i D(, p ) are Hamiltoia w.h.p. for say p > 2 log / ([16]). Sice p = α2 log 2 α log s + m = ω ( ) log(s + m) s + m 9

10 we fid that D(M, V 0 ) is Hamiltoia w.h.p.. Thus we have show that ( ) P r P does ot exted to a Hamilto cycle i D D (V) = o(1). (2) Let Λ V Λ V deote the set of P Λ V which do ot exted to a Hamilto cycle i D D (V). By (2) we have E( Λ V ) = o( Λ V ). Usig Markov s iequality we obtai that Λ V = o( Λ V ) whp. Combied with (1) this gives that Λ V \ Λ V (1 o(1)) (m!) l 1 p s whp. Lastly, to complete the proof of the claim, ote that ay two distict families P, P Λ V \ Λ V yield differet Hamilto cycles ideed, by deletig the vertices of S from the Hamilto cycle it is easy to recover the paths P. This proves the claim. Now to complete the proof of the theorem, let Γ deote the set of (l, s)-partitios V with V 0 = S which satisfy the statemet of the claim. By Markov s iequality we have Γ (1 o(1)) ( s)! whp. (m!) l Sice for distict V, V Γ the Hamilto cycles i D D (V) are all distict, we fid that whp D cotais at least Γ (1 o(1)) (m!) l 1 p s ( s)! (1 o(1)) (m!) l (m!) l 1 p ( s)! = (1 o(1)) p = (1 o(1))!p m! distict Hamilto cycles. The fial equality here holds sice m < /α log gives that m! < e /α ad () s s = (1 + o(1)) sice s = o(/ log ). This completes the proof of the theorem. 4 Packig Hamilto cycles i D(, p) I this sectio we prove Theorem 1.1. The heart of the argumet is cotaied i the followig lemma. Lemma 4.1. Let V = (V 0, V 1,..., V l ) be a (l, s)-partitio of a set X of size = lm + s. Suppose that we select a radom subdigraph F of D (V) as follows: iclude each iterior directed edge of D (V) idepedetly with probability p i ; iclude each exterior directed edge of D (V) idepedetly with probability p ex. The, provided p i = ω(log /m) ad p ex = ω(mp i log /(m + s)), w.h.p. F cotais (1 o(1))mp i edge-disjoit Hamilto cycles. Proof. We begi by exposig the iterior edges of F. For j [l 1] all edges E D (V j, V j+1 ) appear i E F (V j, V j+1 ) idepedetly with probability p i. By igorig the orietatios, we ca view E F (V j, V j+1 ) as a bipartite graph. From Lemma 2.5, sice p i = ω(log m/m) we fid that w.h.p. for all j [l 1] the graph E F (V j, V j+1 ) cotais L := (1 o(1))mp i edge-disjoit perfect matchigs {M j,k } L k=1. For each k [L], combiig the edges i the matchigs {M j,k} j [l 1] gives m directed paths, each directed from V 1 to V l ad coverig l i=1 V i. Let P k,1,..., P k,m deote these paths ad P k = {P k,1,..., P k,m }. Now for each exterior edge e of D (V) choose a value h(e) [L] uiformly at radom, all values chose idepedetly. Now expose the exterior edges of F ad for each i [L] let H i deote the subgraph of F with edge set {e E(F ) : e exterior with h(e) = i}. 10

11 Claim 4.2. For ay k [L], the digraph C k := { e : e is a directed edge of some path i P k } H k cotais a directed Hamilto cycle with probability 1 o(1/). Note that the proof of the lemma follows from the claim, by takig a uio boud over all k [L]. To prove the claim, let M k = {(u k,i, v k,i ) V 1 V l : P k,i is a directed u k,i v k,i path}. By Remark 2.9 it suffices to prove that the auxiliary digraph C k (M k, V 0 ) cotais a directed Hamilto cycle. Note that V (C k (M k, V 0 )) = s + m ad that C k (M k, V 0 ) D(s + m, p ex /L). Sice p ex /L = ω(log /(s + m)), the digraph C k (M k, V 0 ) is Hamiltoia with probability 1 o(1/). By Remark 2.9, this completes the proof of the claim, ad therefore the lemma. The followig lemma allows us to cover the edges of the complete digraph i a reasoably balaced way usig copies of D (V). Lemma 4.3. Suppose that X is a set of size ad that l, m, s N satisfyig = ml + s, with t = ω(l log ), t = ω(( 2 /s 2 ) log ) ad s = o() ad m = o(s). Let V (1),..., V (t) be a collectio of (l, s)-partitios of X chose uiformly ad idepedetly at radom, where V (i) = (V (i) 0,..., V (i) l ). The w.h.p. for each pair u, v X, the directed edge e = uv satisfies: 1. A e = (1 + o(1)) t l where A e := { i [t] : e is a iterior edge of D (V (i) ) }. 2. B e = (1 + o(1)) s2 t 2 where B e := { i [t] : e is a exterior edge of D (V (i) ) }. Proof. Let V (1),..., V (t) be (l, s)-partitios chose uiformly ad idepedetly at radom. Give a fixed directed edge e, the sizes A e ad B e are biomially distributed o a set t with m 2 E( A e ) = (l 1) ( 1) t ad E( B e ) = m2 + 2sm + s(s 1) t. ( 1) Usig that s = o() ad = ml + s this gives that E( A e ) = (1 + o(1)) t l ad usig m = o(s) gives E( B e ) = (1 + o(1)) s2 t. Therefore by Lemma ( Pr A e E( A e ) ) > ae( A e ) 2e a2 E( A e )/3 2e (1+o(1))a2t/3l = o(1/ 2 ). (3) Here ( we used that a 2 t/3l 3 log for a = o(1). Similarly usig that ts 2 / 2 = ω(log ) we fid Pr B e E( B e ) ) > ae( B e ) = o(1/ 2 ). Takig a uio boud over all directed edges gives that w.h.p. 1. ad 2. hold for all e. Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. Let α = α() be some fuctio tedig to ifiity arbitrarily slowly with. Suppose that p α 6 log 4 / ad let l = α 3 log, s = /α 2 log be itegers with = ml + s. Note that this gives m = (1 + o(1))/α 3 log. Additioally set t = α 5 log 3. With these choices, the hypothesis of Lemma 4.3 is satisfied. Let V (1),..., V (t) be a collectio of (l, s)-partitios of X = [], chose so that the coclusios of Lemma 4.3 are satisfied. Therefore A e = (1 + o(1))t/l = (1 + o(1))α 2 log 2 ad B e = (1 + o(1))s 2 t/ 2 = (1 + o(1))α log for every e. 11

12 To begi, wheever we expose the edges of a directed graph D D(, p), we will assig the edges of D amog t edge disjoit subdigraphs D (1),..., D (t). The digraphs D (i) are costructed as follows. For each edge e idepedetly choose a radom value h(e) A e B e where a elemet i A e is selected with probability (1 1/α)/ A e ad a elemet i B e is selected with probability 1/α B e. For each i [t], we take D (i) to be the digraph give by D (i) = {e E(D) : h(e) = i}. We prove that w.h.p. D (i) cotais edge-disjoit Hamilto cycles coverig almost all of its edges. First ote that all edges e of D (V (i) ) appear idepedetly i D (i). If e is a iterior edge the the probability that it appears is p(1 1/α)/ A e (1 o(1))p/α 2 log 2 := p i. Similarly, each exterior edge e i D (V (i) ) appears i D (i) with probability p/α B e (1 o(1))p/α 2 log := p ex. Usig these values, select F as i Lemma 4.1. Also set L = (1 o(1))mp i. Due to mootoicity we coclude that for every i [t] we have Pr(D (i) cotais L edge disjoit Ham. cycles) Pr(F cotais L edge disjoit Ham. cycles). (4) Now we claim with these choices of p i ad p ex the hypothesis of Lemma 4.1 are satisfied. Ideed, usig p α 6 log 4 / gives (1 + o(1))p i = ad so p i = ω(log /m). Similarly we have p α 2 log 2 α4 log 2 = (1 + o(1)) α log m, p p ex = (1 + o(1)) α 2 log = (1 + o(1))p i log = (1 + o(1)) αmp i log, s ad p ex = ω(mp i log /(m+s)). Thus by Lemma 4.1, Pr(F cotais L edge disjoit Ham. cycles) = 1 o(1). Summig over i [t] ad combiig with (4), this proves that w.h.p. D cotais at least (1 o(1))lt = (1 o(1))mp i t = (1 o(1)). s l. pl t.t = (1 o(1))p edge-disjoit Hamilto cycles. 5 Coverig D(, p) with Hamilto cycles I this sectio we prove Theorem 1.2. To begi we first prove the followig lemma. The proof makes use of the max-flow mi-cut theorem ad the itegrality theorem for etwork flows (see Chapter 7 i [37]). Lemma 5.1. Let G = (A, B, E) be a bipartite graph, with A = B = N ad δ(g) d. Suppose that G has the followig properties: For ay X A, Y B with X N 4 ad Y N 4 we have e G(X, Y ) dn 40, For ay X A with X N 4, if e G(X, Y ) 3d X 4 for some Y B the Y 2 X, For ay Y B with Y N 4, if e G(X, Y ) 3d Y 4 for some X A the X 2 Y. The give ay iteger r with r d 80 ad a bipartite graph H o vertex set A B with := (H) r 2, there exists a subgraph G of G which is edge disjoit from H such that G H is r-regular. 12

13 Proof. Give a graph F o V (G) ad a vertex v of G, let d F (v) deote the degree of v i F. By assumptio, we have d H (v) r/2 for all v A B. We wish to fid a subgraph G of G which is edge-disjoit from H so that d G (v) + d H (v) = r for all v V (G). We prove the existece of G by represetig it as a flow i a appropriate etwork. Cosider the followig etwork D o vertex set V (G) {s, t}, with source s ad sik t. For each a A, the edge sa E(D) ad it has capacity r d H (a). For each b B, the edge bt E(D) ad it has capacity r d H (b). Lastly, each edge i E(G \ H) is directed from A to B ad has capacity 1. Usig the itegrality theorem for etwork flows, it is sufficiet to show that there is a flow from s to t of value V = (r d H (a)) = rn d H (a). (5) a A a A By the max-flow mi-cut theorem it is sufficiet to show that D does ot cotai a s t cut of capacity less tha V. To see this, suppose for cotradictio that {s} A s B s ad A t B t {t} forms such a cut, A v A ad B v B for v {s, t}. The capacity of this cut is C = (r d H (a)) + (r d H (b)) + e G\H (A s, B t ). a A t b B s We may assume that A s N/4 or B t N/4. Ideed, otherwise from the statemet of the lemma we have e G (A s, B t ) dn/40 ad C e G\H (A s, B t ) e G (A s, B t ) N dn/40 rn/2 rn V, sice r d/80. We will focus o the case A s N/4 as the case B t N/4 follows from a idetical argumet. Note that sice e G\H (A s, B) (δ(g) ) A s (d ) A s, we fid e G\H (A s, B t ) e G\H (A s, B) e G\H (A s, B s ) (d ) A s e G (A s, B s ). (6) From (6) it follows that if e G (A s, B s ) 3d As 4 the C a A t (r d H (a)) + e G\H (A s, B t ) a A t (r d H (a)) + (d 3d 4 ) A s a A t (r d H (a)) + r A s V, where the secod last iequality holds sice d/4 2r + r ad the last iequality holds by (5). If e G (A s, B s ) 3d As 4, sice A s A /4, by the hypothesis of the lemma we have B s 2 A s. But the, sice r/2 we have C (r d H (a)) + (r d H (b)) (r d H (a)) + B s (r ) a A t b B s a A t a A t (r d H (a)) + 2 A s r 2 a A(r d H (a)) = V. This covers all cases, ad completes the proof. 13

14 We ow prove a coverig versio of Lemma 4.1. I our proof of Theorem 1.2 we will agai break D D(, p) ito may sub-digraphs which are distributed similarly to F from Lemma 4.1. However there will be some small fluctuatio i the edge probabilities of edges i these sub-digraphs. The slightly uusual phrasig of the ext lemma is iteded to allow for these fluctuatios. Lemma 5.2. Let V = (V 0, V 1,..., V l ) be a (l, s)-partitio of a set X of size = ml + s. Choose a radom subdigraph F of D (V) as follows: iclude each iterior edge e from D (V) idepedetly with probability q e (1 ± o(1))p i ; iclude each exterior edge from D (V) idepedetly with probability at least p ex. The, provided p i = ω(log /m) ad p ex = ω(log /(m + s)), with probability 1 o(1/ 2 ) there are (1 + o(1))mp i directed Hamilto cycles i F which cover all iterior edges of F D (V). Proof. We begi by exposig the iterior edges of F. For ay j [l 1], all of edges E D (V j, V j+1 ) appear i E F (V j, V j+1 ) idepedetly with probability at least (1 o(1))p i. For ay j [l 1], let F j be the subdigraph of F cosists of the vertices V j V j+1 ad the edges i E F (V j, V j+1 )). We agai view F j as a bipartite graph, simply by igorig the orietatios. As i Lemma 4.1, with probability 1 o(1/ 2 ) for each j [l 1] we ca fid L = (1 o(1))mp i edge-disjoit perfect matchigs i E F (V j, V j+1 ), which we deote by {M j,k } L k=1. Now remove the edges of these matchigs from E F (V j, V j+1 ) ad let H j deote the remaiig subdigraph. Sice p i = ω(log /m) ad q (1 + o(1))p i, by Cheroff s iequality, with probability 1 o(1/ 2 ) every vertex u V j ad v V j+1 satisfies (1 + o(1))mp i d + F j (u), d F j (v) (1 + o(1))mp i. Therefore with probability 1 o(1/ 2 ), for all j [l 1], such u ad v satisfy d + H j (u) = o(mp i ) ad d H j (v) = o(mp i ). (7) Now give X V j ad Y V j+1 we also have E ( e Fj (X, Y ) ) = (1 ± o(1)) X Y p i. Cheroff s iequality therefore shows that P r ( e Fj (X, Y ) (1 ± o(1)) X Y p i > t ) e t2 /4 X Y p i. Usig this boud it is easy to check that the followig holds: with probability 1 ω(1), for all j [l 1] the hypothesis of Lemma 5.1 are satisfied by the bipartite graph F j, takig d = (1 o(1))mp i ad N = m. Settig r = max j [l 1] {2 (H j )}, from (7) we have r d for all j [l 1]. Therefore by Lemma 5.1, with probability 1 ω(1), for all j [l 1] the graph F j cotais a r-regular subgraph G j which icludes all edges of H j. Now by Hall s theorem, for each j [l 1] the digraph G j ca be decomposed ito r edgedisjoit perfect matchigs, which we deote by {M j,k } L+r k=l+1. Combied with the matchigs at the begiig of the proof, we have show that with probability 1 o(1/ 2 ), for each j [l 1] there are perfect matchigs {M j,k } L+r k=1 which cover all iterior directed edges of F. By combiig the edges {M j,k } l 1 j=1 for each k [L + r], we get m directed paths, each directed from V 1 to V l ad coverig l i=1 V i. Let P k,1,..., P k,m deote these paths ad P k = {P k,1,..., P k,m }. I particular these paths cover all iterior edges of F. 14

15 Now to complete the proof we expose the exterior edges of F ad use them to complete each P k ito a directed Hamilto cycle as i the proof of Lemma 4.1. For each k [L + r] let M k = {(u k,i, v k,i ) V 1 V l : P k,i is a directed u k,i v k,i path}. Now V (F (M k, V 0 )) = s + m ad as i Lemma 4.1, F (M k, V 0 ) D(s + m, p ex ). Sice p ex = ω(log /(s + m)), the digraph F (M k, V 0 ) is Hamiltoia with probability 1 o(1/ 3 ). By Remark 2.9, this shows that with probability 1 o(1/ 2 ), for all k [L + r] the digraph F cotais a directed Hamilto cycle cotaiig all edges of the paths i P k. As these paths cover all iterior edges of F, this completes the proof of the lemma. Proof of Theorem 1.2. The proof follows a similar argumet to that of Theorem 1.1. Let α = α() be some fuctio tedig arbitrarily slowly to ifiity with ad let p α 4 log 2 /. Let = ml+s where l = α, s = /α ad t = α 2 log. Note that m = (1 + o(1))/α. Sice t = ω(l log ) we ca take V (1),..., V (t) to be a collectio of (l, s)-partitios of X = [] as give by Lemma 4.3. To begi, wheever we expose the edges of a directed graph D D(, p), we will assig the edges amog t sub-digraphs D (1),..., D (t). The digraphs D (i) are costructed as follows. Let A e := {i [t] : e is iterior i D (V (i) )}. By Lemma 4.3, w.h.p. for each edge e we have A e = (1 + o(1))t/l = (1 + o(1)α log. Idepedetly for each edge e choose a value h(e) A e uiformly at radom. For each i [t], let the digraph D (i) cotai the edges {e E(D) : h(e) = i}. Furthermore, adjoi all edges of D which occur as a exterior edge of D (V (i) ) to D (i). We will prove that w.h.p. D (i) cotais directed Hamilto cycles coverig all the edges of D D (V (i) ). First ote that all edges e of D (V (i) ) appear idepedetly i D (i). If e is a iterior edge the the probability that it appears is p/ A e = (1 ± o(1))p/α log = p i. We see that each iterior edge of D (V (i) ) appears i D (i) idepedetly with probability betwee (1 o(1))p i ad (1 + o(1))p i. Also, each exterior edge e i D (V (i) ) appears i D (i) with probability p ex := p. Now we have p ex = p = ω( log m+s ). We also have p i = (1 + o(1)) pl t Lemma 5.2 we obtai α2 log = α log m, so p i = ω( log m ). Thus by Pr(D (i) has (1 + o(1))mp i directed Hamilto cycles coverig its iterior edges) (8) Summig (8) over i [t], this proves that w.h.p. D cotais (1+o(1))mp i t = (1+o(1))p Hamilto cycles coverig the iterior edges of D (i) for all i [t]. Sice each edge of D occurs as a iterior edge of D (i) for some i [t], this completes the proof of the theorem. 6 Packig Hamilto cycles i pseudo-radom directed graphs 6.1 Pseudo-radom digraphs ad Hamiltoicity Defiitio 6.1. A directed graph D o vertices is called (, λ, p)-pseudo-radom if the followig hold: (P1) (1 λ)p d + D (v), d D (v) (1 + λ)p for every v V (D); (P2) For every X V (D) of size X 4 log8 p we have e D (X) (1 λ) X log 8.02 ; 15

16 (P3) For every two disjoit subsets X, Y V (D) of sizes X, Y log1.1 p we have e D (X, Y ) = (1 ± λ) X Y p. The followig theorem of Ferber, Neadov, Noever, Peter ad Skorić [13] gives a sufficiet coditio for pseudo-radom digraph to be Hamiltoia. Theorem 6.2. (Theorem 3.2, [13]) Let 0 < λ < 1/10. The for p = ω( log8 ) the followig holds. Let D be a directed graph with the followig properties: (P1) (1 λ)p d + D (v), d D (v) (1 + λ)p for every v V (D); (P2)* for every X V (D) of size X log2 p we have e D (X) X log 2.1 ; (P3)* for every two disjoit subsets X, Y V (D) of sizes X, Y log1.1 p (1 + λ) X Y p. we have e D (X, Y ) The D cotais a Hamilto cycle. 6.2 Properties of pseudo-radom graphs The followig lemmas will be useful i the proof of Theorem 1.4. We have deferred the proofs to the Appedix. I these lemmas we assume that p = ω(log 14 /), p = p/ log 6, s = /αp ad m = s/ log. Lemma 6.3. Let D be a (, λ, p)-pseudo-radom digraph with 0 < λ < 1 ad p = ω(log 14 /). We first select a radom subdigraph C of D by icludig edges idepedetly with probability q (1 ± o(1))p /p. The select a (l, s)-partitio of V (D) give by V = (V 0, V 1,..., V l ) uiformly at radom, with V 0 = s ad = ml + s. The with probability 1 o(1/) the followig holds: for every collectio M of m disjoit pairs from V 1 V l, the radom digraph F 0 = C(M, V 0 ) satisfies the followig properties: (A) (1 3λ)(s + m)p d + F 0 (v), d F 0 (v) (1 + 3λ)(s + m)p for every v V (F 0 ), (B) we have e F0 (X) X log 2.1 for every X V (F 0 ) of size X log2 (s+m) p, (C) for every two disjoit subsets X, Y V (F 0 ) of sizes X, Y log1.1 (s+m) p, we have e F0 (X, Y ) (1 + 2λ) X Y p. Lemma 6.4. Let D be (, λ, p) pseudo-radom digraph with 0 < λ < 1/4 ad p = ω(log 14 /). Suppose that V (D) = V 0 V 1 V l is a radom (l, s)-partitio of V (D) with V 0 = s ad = ml + s ad let F be the graph obtaied from D by keepig every iterior edge with probability p i = 1/(αl log ). The with probability 1 o(1/), for every j [l 1] the directed subgraph F j = E F (V j, V j+1 ) cotais (1 4λ)mp p i edge disjoit perfect matchigs. 16

17 6.3 Proof of Theorem 1.4 To prove Theorem 1.4 we use the followig lemma (the aalogue of Lemma 4.1) about the existece of may edge disjoit Hamilto cycles i special pseudo-radom directed graphs. Lemma 6.5. Let V = (V 0, V 1,..., V l ) be a (l, s)-partitio of a set X of size = lm + s, chose uiformly ad idepedetly at radom. Let D be a (, λ, p) pseudo-radom graph o the vertex set X, with p = ω(log 14 /), ad 0 < λ < 1/100. Suppose that we select a radom subdigraph F of D(V) as follows: iclude each iterior edge of D(V) idepedetly with probability p i ; iclude each exterior edge of D(V) idepedetly with probability p ex. The, provided p i = (1 o(1)) 1/(αl log ) ad p ex = 2 /(α 2 s 2 l 2 log ), where p = p/ log 6, s = /αp, m = s/ log ad α = α() is some fuctio tedig arbitrarily slowly to ifiity with, F cotais (1 o(1))(1 4λ)mpp i edge-disjoit Hamilto cycles with probability 1 o(1/). Proof. To begi, look at the iterior edges of F. For j [l 1] all edges of E D (V j, V j+1 ) appear i F idepedetly with probability p i. Lemma 6.4 therefore gives that with probability 1 o(1/t), E F (V j, V j+1 ) cotais L := (1 4λ)mpp i edge-disjoit perfect matchigs {M j,k } L k=1 for all j [l 1]. For each k [L], takig the uio of the edges i the matchigs l 1 j=1 M j,k gives m directed paths, each directed from V 1 to V l ad coverig l i=1 V i. Let P k,1,..., P k,m deote these paths ad P k = {P k,1,..., P k,m }. Now assig to each exterior edge e of D(V) a value h(e) [L] chose uiformly at radom, all values chose idepedetly. Look at the exterior edges of F ad for each i [L] let H i deote the subgraph of F with edge set {e E(F ) : e exterior with h(e) = i}. Claim 6.6. For ay k [L], the digraph C k := { e : e is a directed edge of some path is P k } H k cotais a directed Hamilto cycle with probability 1 o(1/). Note the proof of the lemma immediately follows from the claim, summig over k [L]. To prove the claim, let M k = {(u k,i, v k,i ) V 1 V l : P k,i is a u k,i v k,i directed path}. By Remark 2.9 it suffices to prove that the auxiliary digraph C k (M k, V 0 ) cotais a directed Hamilto cycle. Now ote that V (C k (M k, V 0 )) = s+m. We ow wish to prove that with probability 1 o(1/) every C k (M k, V 0 ) is Hamiltoia. Observe that each C k (M k, V 0 ) was created from F (M k, V 0 ) by keepig each edge e with probability at least p ex / L = 2 α 2 s 2 l 2 log 1 (1 o(1))mpp i. Usig that p i = (1 o(1))1/(αl log ), p = /αs 2 ad that ml = (1 o(1)) gives that p ex / L = (1 o(1))p /p. By applyig Lemma 6.3, we see that C k (M k, V 0 ) satisfies properties (A), (B) ad (C) with probability 1 o(1/). But (A), (B) ad (C) give properties (P 1), P (2) ad P (3) from Theorem 6.2, takig p i place of p. Sice p = (1 o(1))p/ log 6 = ω(log 8 /), by Theorem 6.2 ay such C k (M k, V 0 ) are Hamiltoia. But if C k (M k, V 0 ) is Hamitoia, the so is C k. Thus, this proves that C k is Hamiltoia with probability 1 o(1/). 17

18 Now we are ready to prove Theorem 1.4. Proof of Theorem 1.4. Let α = α() be some fuctio tedig arbitrarily slowly to ifiity with. Let = ml + s where m = s/ log ad s = /αp. Let V (1),..., V (t) be a collectio of (l, s)-partitios of X = [], where V (i) = (V (i) 0,..., V (i) l ), chose uiformly ad idepedetly at radom ad t = αl 2 log. We will assig the edges of D amog t edge disjoit subdigraphs D (1),..., D (t) such that each D (i) preserves some pseudo-radom properties. The digraphs D (i) are costructed as follows. Let A e := {i [t] : e is iterior i D(V (i) )}; B e := {i [t] : e is exterior i D(V (i) )}. By Lemma 4.3, w.h.p. for each edge e we have A e = (1 + o(1)) t l ad B e = ((1 + o(1)) s2 t 2. We will ow show that there exists a fuctio f, f(e) A e B e, such that if D (i) is the digraph give by D (i) = {e : f(e) = i}, the D (i) cotais L := (1 o(1))p/t directed Hamilto cycles. Clearly, this will complete the proof. For each edge e choose a radom value f(e) A e B e where each elemet i A e is selected with probability (1 1/α)/ A e ad each elemet i B e is selected with probability 1/α B e. For each i [t], we take D (i) to be the digraph give by D (i) = {e : f(e) = i}. First ote that all edges e of D(V (i) ) appear idepedetly i D (i). If e E(D) is a iterior edge the the probability that it appears is (1 1/α)/ A e (1 o(1))l/t = (1 o(1))1/αl log := p i, sice t = αl 2 log. Similarly, each exterior edge e i D D(V (i) ) appears i D (i) with probability 1/α B e (1 o(1)) 2 /αts 2 = (1 o(1)) 2 /α 2 l 2 s 2 log := p ex. Now ote that the coditios of Lemma 6.5 are satisfied with these values (with D (i) i place of F ), so with probability 1 o(1/), D (i) cotais L = (1 o(1))(1 4λ)mpp i edge disjoit Hamilto cycles. Therefore with probability 1 o(1), D (i) cotais L edge disjoit Hamilto cycles for each i [t]. Fix a choice of V (1),..., V (t) ad f such that this holds. Usig that p i t = l this gives that D cotais (1 o(1))lt = (1 4λ o(1))mpp i t = (1 4λ o(1))mpl (1 5λ)p edge-disjoit Hamilto cycles, as required. Ackowledgmet. The authors would like to thak the referee of the paper for his careful readig ad may helpful remarks. Refereces [1] N. Alo ad J. H. Specer. The probabilistic method. Joh Wiley & Sos, [2] S. Be-Shimo, M. Krivelevich, ad B. Sudakov. O the resiliece of Hamiltoicity ad optimal packig of Hamilto cycles i radom graphs. SIAM Joural o Discrete Mathematics, 25(3): , [3] B. Bollobás. The evolutio of radom graphs. Trasactios of the America Mathematical Society, 286(1): , [4] B. Bollobás ad A. M. Frieze. O matchigs ad Hamiltoia cycles i radom graphs. North- Hollad Mathematics Studies, 118:23 46, [5] B. Csaba, D. Küh, A. Lo, D. Osthus, ad A. Treglow. Proof of the 1-factorizatio ad Hamilto decompositio cojectures. Memoirs of the America Mathematical Society, to appear. 18

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