Fast Strategies In Maker-Breaker Games Played on Random Boards

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1 Fast Strategies I Maker-Breaker Games Played o Radom Boards Deis Clemes Asaf Ferber Michael Krivelevich Aita Liebeau March 15, 2012 Abstract I this paper we aalyze classical Maker-Breaker games played o the edge set of a sparse radom board G G,p. We cosider the Hamiltoicity game, the perfect matchig game ad the k-coectivity game. We prove that for p polylog/, the board G G,p is typically such that Maker ca wi these games asymptotically as fast as possible, i.e. withi + o, /2 + o ad k/2 + o moves respectively. 1 Itroductio Let X be ay fiite set ad let F 2 X be a family of subsets. Usually, X is called the board, whereas F is referred to as the family of wiig sets. I the a : b Maker-Breaker game X, F also kow as a weak game, two players called Maker ad Breaker play i rouds. I every roud Maker claims a previously uclaimed elemets of the board X ad Breaker respods by claimig b previously uclaimed elemets of the board. Maker wis as soo as he fully claims all elemets of some F F. If Maker does ot fully claim ay wiig set by the time all board elemets are claimed, the Breaker wis the game. The most basic case is a = b = 1, the so called ubiased game. Notice that beig the first player is ever a disadvatage i a Maker-Breaker game. Therefore, i order to prove that Maker ca wi some Maker-Breaker game as the first or the secod player it is eough to prove that he ca wi this game as a secod player. Hece, we will always assume that Maker is the secod player to move. It is atural to play Maker-Breaker games o the edge set of a graph G = V, E with V =. I this case, X = E. I the coectivity game, Maker wis if ad oly if his edges cotai Departmet of Mathematics ad Computer Sciece, Freie Uiversität Berli, Germay. d.clemes@fu-berli.de. Research supported by DFG, project SZ 261/1-1. School of Mathematical Scieces, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv, 69978, Israel. ferberas@post.tau.ac.il School of Mathematical Scieces, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv, 69978, Israel. krivelev@post.tau.ac.il. Research supported i part by USA-Israel BSF grat , by grat 1063/08 from the Israel Sciece Foudatio, ad by a Pazy memorial award. Departmet of Mathematics ad Computer Sciece, Freie Uiversität Berli, Germay. liebeau@math.fu-berli.de. Research supported by DFG withi the graduate school Berli Mathematical School. 1

2 a spaig tree. I the perfect matchig game M G the wiig sets are all sets of /2 idepedet edges of G. Note that if is odd, the such a matchig covers all vertices of G but oe. I the Hamiltoicity game H G the wiig sets are all edge sets of Hamilto cycles of G. Give a positive iteger k, i the k-coectivity game C k G the wiig sets are all edge sets of k-coected subgraphs of G. Maker-Breaker games played o the edge set of the complete graph K are well studied. I this case, it is easy to see ad also follows from [16] that for every 4, Maker ca wi the ubiased coectivity game i 1 moves, which is clearly the best possible. It was proved i [12] that Maker ca wi the ubiased perfect matchig game o K withi /2 + 1 moves which is clearly the best possible, the ubiased Hamiltoicity game withi + 2 moves ad the ubiased k-coectivity game withi k/2 + o moves. I [13], it was show that Maker ca wi the ubiased Hamiltoicity game o K withi + 1 moves which is clearly the best possible ad recetly it was proved see [8] that Maker ca wi the ubiased k-coectivity game withi k/2 + 1 moves which is clearly the best possible. It follows from all these results that may atural games played o the edge set of the complete graph K are drastically i favor of Maker. Hece, it is atural to try to make his life a bit harder ad to play o differet types of boards or to limit his umber of moves. I this paper we are maily iterested i the followig two questios. i Give a sparse board G = V, E, ca Maker wi the game played o this board? ii How fast ca Maker wi this game? I [17] it was suggested to play Maker-Breaker games o the edge set of a radom graph G G,p ad some games were examied such as the perfect matchig game, the Hamiltoicity game, the coectivity game ad the k-clique game. Later o, i [5], it was proved that the edge set of G G,p with p = 1 + o1 l is typically such that Maker has a strategy to wi the ubiased perfect matchig game, the Hamiltoicity game ad the k-coectivity game. This is best possible sice p = l is the threshold probability for the property of G,p havig a isolated vertex. Moreover, the proof i [5] is of a hittig-time type. That meas, i the radom graph process, i.e. whe addig oe ew edge radomly every time, typically at the momet the graph reaches the eeded miimum degree for wiig the desired game, Maker ideed ca wi this game. For example, at the first time the graph process achieves miimum degree 2 the board is typically such that Maker wis the perfect matchig game. Aother type of games is the followig. Let X be ay fiite set ad let F 2 X be a family of subsets. I the strog game X, F, two players called Red ad Blue, take turs i claimig oe previously uclaimed elemet of X, with Red goig first. The wier of this game is the first player to fully claim all the elemets of some F F. If o oe wis by the time all the elemets of X are claimed, the the game eds i a draw. For example, the classic Tic-Tac-Toe is such a game. It is well kow from classic Game Theory, that for every strog game X, F, either Red has a wiig strategy or Blue has a drawig strategy. For certai games, a hypergraph colorig argumet ca be used to prove that a draw is impossible ad thus these games are wo by Red. However, these argumets are purely existetial. That is, 2

3 eve if it is kow that Red has a wiig strategy for some strog game X, F, it might be very hard to describe such a strategy explicitly. Usig fast strategies for Maker i certai games, explicit strategies for Red were give for games such as the perfect matchig game, the Hamiltoicity game ad the k-coectivity game played o the edge set of K see [8] ad [7]. This provides substatial motivatio for studyig fast wiig strategies i Maker-Breaker games. Regardig the strog game played o G G,p, othig is kow yet. Hece, as a first step for fidig explicit strategies for Red i the strog game played o a radom board, it is atural to look for fast wiig strategies for Maker i the aalogous games. Therefore the followig questio is quite atural. Questios : Give p = p, how fast ca Maker wi the perfect matchig, the Hamiltoicity ad the k-coectivity games played o the edge set of a radom board G G,p? I this paper we resolve these questios for a wide rage of the values of p = p. We prove the followig theorems: Theorem 1.1 Let b 1 be a iteger, let K > 12, p = lk, ad let G G,p. The a.a.s. G is such that i the 1 : b weak game M G, Maker has a strategy to wi withi 2 + o moves. Theorem 1.2 Let b 1 be a iteger, let K > 100, p = lk, ad let G G,p. The a.a.s. G is such that i the 1 : b weak game H G, Maker has a strategy to wi withi + o moves. Theorem 1.3 Let b 1, k 2 be two itegers, let K > 100, p = lk, ad let G G,p. The a.a.s. G is such that i the 1 : b weak game CG, k Maker has a strategy to wi withi + o moves. k 2 Due to obvious mootoicity the results are valid for ay p = p larger tha stated i the theorems above. For the sake of simplicity ad clarity of presetatio, we do ot make a particular effort to optimize the costats obtaied i our proofs. We do ot believe that the order of magitude we assume for p i the above theorems is optimal. We also omit floor ad ceilig sigs wheever these are ot crucial. Most of our results are asymptotic i ature ad wheever ecessary we assume that is sufficietly large. The remaiig part of the paper is orgaized as follows. First, we itroduce the ecessary otatio. I Sectio 2, we assemble several results that we eed. We give some basic results of positioal games i Sectio 2.1, of graph theory i Sectio 2.2, ad about G,p i Sectio 2.3. The strategy of Maker i each of the three games icludes buildig a suitable expader o a subgraph, which the cotais the desired structure. We therefore iclude results about expaders i Sectio 2.4. We prove Theorem 1.1, 1.2 ad 1.3 i Sectios 3, 4 ad 5, respectively. Fially, i Sectio 6 we pose some ope problems coected to our results. 3

4 1.1 Notatio ad termiology Our graph-theoretic otatio is stadard ad follows that of [18]. I particular, we use the followig. For a graph G, let V G ad EG deote its sets of vertices ad edges, respectively. Let S, T V G be subsets. Let G[S] deote the subgraph of G, iduced o the vertices of S, ad let ES = EG[S]. Further, let ES, T := {st EG : s S, t T }, ad let NS := {v V : s S s.t. vs EG} deote the eighborhood of S. For a edge e EG we deote by G e the graph with vertex set V G ad edge set EG \ {e}. Assume that some Maker-Breaker game, played o the edge set of some graph G, is i progress. At ay give momet durig the game, we deote the graph formed by Maker s edges by M, ad the graph formed by Breaker s edges by B. At ay poit durig the game, the edges of F := G \ M B are called free edges. For subsets S, T V, let N M S := {v V : s S s.t. vs EM}, E M S := ES EM, ad E M S, T := ES, T EM. Further, for v V, let d M v, S = E M {v}, S, ad d M v := d M v, V. Aalogously, we defie N B S, N F S, E B S, E F S, etc. 2 Auxiliary results I this sectio we preset some auxiliary results that will be used throughout the paper. First, 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], [14]. Lemma 2.1 If X Bi, p, the Pr[X < 1 ap] < exp a2 p 2 for every a > 0. Pr[X > 1 + ap] < exp p 3 for every a 1. Lemma 2.2 Let X Bi, p, µ = EX ad k 7µ, the PrX k e k. 2.1 Basic positioal games results The followig fudametal theorem, due to Beck [2], is a useful sufficiet coditio for Breaker s wi i the a : b game X, F. It will be used extesively throughout the paper. Theorem 2.3 Let X be a fiite set ad let F 2 X. If F F 1 + b F /a < 1 1+b, the Breaker as the first or secod player has a wiig strategy for the a : b game X, F. While Theorem 2.3 is useful i provig that Breaker wis a certai game, it does ot show that he wis this game quickly. The followig lemma is helpful i this respect. 4

5 Lemma 2.4 Trick of fake moves Let X be a fiite set ad let F 2 X. Let b < b be positive itegers. If Maker has a wiig strategy for the 1 : b game X, F, the he has a strategy to wi the 1 : b game X, F withi 1 + X /b + 1 moves. The mai idea of the proof of Lemma 2.4 is that, i every move of the 1 : b game X, F, Maker i his mid gives Breaker b b additioal board elemets. The straightforward details ca be foud i [3]. We will also use a variat of the classical Box Game first itroduced by Chvátal ad Erdős i [6]. The Box Game with resets rboxm, b, first studied i [9], is played by two players, called BoxMaker ad BoxBreaker. They play o a hypergraph H = {A 1,..., A m }, where the sets A i are pairwise disjoit. BoxMaker claims q elemets of m i=1 A i per tur, ad the BoxBreaker respods by resettig oe of BoxMaker s boxes, that is, by deletig all of BoxMaker s elemets from the chose hyperedge A i. Note that the chose box does ot leave the game. At every poit durig the game, ad for every 1 i m, we defie the weight of box A i to be the umber of BoxMaker s elemets that are curretly i A i, that is, the umber of elemets of A i that were claimed by BoxMaker ad have ot bee deleted yet by BoxBreaker. Theorem 2.5 Theorem 2.3 i [9] For every iteger k 1, BoxBreaker has a strategy for the game rboxm, b which esures that, at ay poit durig the first k rouds of the game, every box A i has weight at most b1 + lm + k. We will use this theorem as a strategy for Maker to obtai some miimum degree i his graph. To that ed, let G = V, E be some graph, ad let V 1, V 2 V be arbitrary subsets. By 1 : b DegV 1, V 2 we deote the 1 : b positioal game where Maker tries to get a large degree d M v, V 2 for every v V 1. Occasioally, we shall simply refer back to this as the degree game. The followig is a immediate coclusio of Theorem 2.5. Claim 2.6 The degree game Let G = V, E be a graph o V = vertices, V 1, V 2 V, ad let b be a iteger. The, i the 1 : b DegV 1, V 2 game, Maker ca esure that d B v, V 2 4b l d M v, V for every vertex v V 1. Proof Maker preteds he is BoxBreaker ad that he is playig the rbox, 2b game with the boxes E{v}, Nv, V 2, v V 1. Notice that these boxes are ot ecessarily disjoit, sice we did ot require V 1 ad V 2 to be disjoit. However, ay edge belogs to at most two of these boxes. So BoxBreaker ca preted that the boxes are disjoit ad that BoxMaker claims 2b elemets i every move usig the Trick of fake moves. Now, accordig to Theorem 2.5, BoxBreaker ca esure that at ay poit durig the first k rouds of the game, every box has weight at most 2b1 + l + k. Hece, at the ed of the game every box has weight at most 4b l. So, for every vertex v V 1, Maker BoxBreaker has claimed at least oe icidet edge of v for every 4b l icidet edges Breaker BoxMaker has claimed. Hece d B v, V 2 4b l d M v, V Geeral graph theory results We will use the followig graph which was itroduced i [8]. Let k 2 ad 3k 1 be positive itegers such that k 1. Let m := k 1. Let C 1,..., C k 1 be k 1 pairwise vertex 5

6 disjoit cycles, each of legth m. For every 1 i < j k 1 let P ij be a perfect matchig i the bipartite graph V C i V C j, {uv : u V C i, v V C j }. Let G k be the family of all graphs G k = V k, E k where V k = k 1 i=1 V C k 1 i ad E k = i=1 EC i 1 i<j k 1 P ij. For the coveiece of the reader we prove the followig lemma. Lemma 2.7 For all itegers k 2 ad 3k 1 such that k 1, every G k G k is k-regular ad k-vertex-coected. Proof For k = 2, the lemma is trivial. So assume k 3. It is obvious that G k is k- regular. Let S V k be a arbitrary set of size at most k 1. We will prove that G k \ S is coected. Assume first that there exists some 1 i k 1 for which S V C i =. The G k \ S C i = C i is coected ad V C i is a domiatig set of G k \ S. It follows that G k \ S is coected. Assume the that S V C i = 1 for every 1 i k 1. Hece, G k \ S C i is a path o k 1 vertices for every 1 i k 1. Sice k 1 2 ad S V C i = 1 for every 1 i k 1 hold by the assumptio, it follows that there is at least oe edge betwee G k \S C i ad G k \S C j for every 1 i < j k 1. It follows that G k \S is coected. The followig lemma shows that if a directed graph satisfies some pseudo-radom properties the it cotais a log directed path. We will use it i the proof of Theorem 1.2. Lemma 2.8 Lemma 4.4 [4] Let m be a iteger ad let D = V, E be a orieted graph with the followig property: There exists a edge from S to T betwee ay two disjoit sets S, T V such that S = T = m. The, D cotais a directed path of legth at least V 2m + 1. The ext lemma provides a sufficiet Hall-type coditio for a bipartite graph to cotai a perfect matchig. Lemma 2.9 Let G = U 1 U 2, E be a bipartite graph with U 1 = U 2 =. Let r /2 be a iteger such that: B1 For i {1, 2} ad every X U i of size X r, NX X. B2 For every X U 1 ad Y U 2 with X = Y = r, EX, Y > 0. The G has a perfect matchig. Proof I order to prove that G admits a perfect matchig we will prove that G satisfies Hall s coditio, that is, NX X for every X U 1 see e.g. [18]. We show that for every i {1, 2} ad for every X U i of size X /2 we have that NX X. This readily implies Hall s coditio. We distiguish two cases: Case 1: If X r, the by i we have that NX X ad we are doe. Case 2: If r < X /2. By ii we have that NX r /2 X. 6

7 2.3 Properties of G,p This subsectio specifies properties A1-A3 that a graph G G,p fulfils a.a.s. It turs out that these properties are all we eed to prove our mai theorems. So i fact, we could stregthe them to hold for ay graph G that has suitable pseudo-radom properties. Lemma 2.10 Let K 2 ad let G G,p with p = l K /. Further, let α R such that 1 α < K, ad let f = f be some fuctio that satisfies 1 f = Ol l 3. The, a.a.s. A1 δg = Θ l K ad G = Θ l K. A2 For every subset U V, EU max{100 U l, 100 U 2 p}. A3 For ay two disjoit subsets U, W V with U = W = f 1 l α, EU, W = Ω f 2 l K 2α ad EU = Ω f 2 l K 2α. Proof To prove A1, let v V. Sice d G v Bi 1, p we coclude Ed G v = 1p = l K 1 o1. Hece, by Lemma 2.1, Pr d G v 1 1/2 l K exp lk 1 o1 8 Now, by the uio boud argumet we coclude that Pr v V : d G v lk o1/ = o1. 2 Similarly, by Lemma 2.1, we obtai Pr v V : d G v 2 l K = o1. To prove A2, let U V be a fixed subset of size t := U. The E EU t 2 p < 100t 2 p max{100t l, 100t 2 p}. Thus, by Lemma 2.2, Pr EU max{100t l, 100t 2 p} exp exp = o1/. max{100t l, 100t 2 p} 100t l. 7

8 It follows that Pr U V : EU > max{100 U l, 100 U 2 p} exp 100t l t t=1 exp t l 100t l t=1 t=1 exp 99 l = exp 98 l = o1. For A3, let U, W V be two disjoit subsets such that U = W = f 1 l α. Note that EU, W is biomially distributed with expectatio µ := f 2 l K 2α. So by Lemma 2.1 we have that Pr EU, W µ 2 exp µ 8. Applyig uio boud we get that Pr disjoit U, W V : U = W = f l α ad EU, W µ 2 2 exp µ l α 8 2 exp l α α l l + 1 lk 2α 8f 2 sice K > α. = o1, Fially, sice EU Bi U EU = Ω f 2 l K 2α for all U with U = f 1 l α. 2, p, it follows aalogously that a.a.s. 2.4 Expaders Defiitio 2.11 Let G = V, E be a graph with V =. Let R := R ad c := c be two positive itegers. We say that the graph G is a R, c-expader if it satisfies the followig two properties: E1 For every subset X V with X R, NX \ X c X. E2 EX, Y > 0 for every two disjoit subsets X, Y V of size X = Y = R. Recall that a graph G = V, E is called Hamilto-coected if for every x, y V, the graph G cotais a Hamilto path with x ad y as its edpoits. The followig sufficiet coditio for a graph to be Hamilto-coected was itroduced i [10]. Theorem 2.12 Let be sufficietly large, ad let G = V, E be a expader o vertices. The G is Hamilto-coected. 8 / l, l l -

9 That is, by esurig expader properties locally, we ca eforce a Hamilto cycle i the whole graph global property. The followig theorem lies i the heart of all of our proofs. It says that i a subgraph of G of subliear order where certai properties hold Maker is able to build a suitable expader fast, that is i o moves. Theorem 2.13 Let b be a iteger, K > 12, let be a sufficietly large iteger ad let p = l K /. Let H = V H, E H be a graph o V H = Θ / l 4 vertices ad let M ad F be two edge disjoit subgraphs of H, where E M already belogs to Maker ad F cosists of free edges. Assume that the followig properties hold: 1 E M E F = E H. 2 There exist a partitio V H = A 1 A 2 ad a costat c 1 > 0 such that A 1 = {v V H : d M v c 1 l K 6 }, A 2 = {v V H : d F v c 1 l K 4 }, ad A 1 = O l 6 K. 3 For ay two disjoit subsets U, W V H of size U = W = l 5 l l 3, E H U, W = Ω l K 10 l l 6. 4 For every subset U V H, E H U max{100 U l, 100 U 2 p}. The, for every c l l V H ad R = V H / l V H, i the 1 : b Maker-Breaker game played o E H, Maker has a strategy to build a R, c-expader withi o moves. Before we prove this theorem we eed a auxiliary result. Cosider a graph H with edgedisjoit subgraphs M ad F such that the assumptios of Theorem 2.13 hold. Give a subgraph H 1 = V H, E 1 of H, we deote M 1 ad F 1 to be the restrictios of M ad F respectively to the subgraph H 1. The followig lemma says that i H we ca fid a sparse subgraph with suitable properties that will guaratee Maker s wi i the expader game. Lemma 2.14 Uder the assumptios of Theorem 2.13 there exists a subgraph H 1 = V H, E 1 of H with the followig properties: i For every v A 2, d F1 v = Ωl 3. ii For ay two disjoit subsets U, W V H such that U = W = E 1 U, W = Ω. l 3 l l 6 iii For every U V H, E 1 U max{1000 U l, 1000 U 2 l 7 /}. iv E 1 = o. l 5 l l 3, 9

10 Note that all size parameters i this lemma do ot deped o K aymore. We wat to stress that this is crucial for obtaiig E 1 = o. Proof Let ρ = l 7 K. Pick every edge of H to be a edge of H 1 with probability ρ idepedetly of all other choices. Let s :=. The properties i-iv will all l 5 l l 3 be prove by idetifyig the correct biomial distributio ad by applyig Cheroff- ad uio-boud-type argumets. To prove property i, otice that for every v A 2 the degree of v i F 1 is biomially distributed, that is, d F1 v Bid F v, ρ with mea Ed F1 c 1 l 3. Therefore, by Lemma 2.1 we have Pr d F1 v 1 2 c 1 l 3 exp 1 8 c 1 l 3. Hece, by the uio boud we coclude Pr v A 2 : d F1 v 1 2 c 1 l 3 = o1. For property ii, let c 2 > 0 be such that E H U, W c 2 l K 10 /l l 6 for every two disjoit subsets U, W V H with U = W = s. This c 2 clearly exists by assumptio 3. Let U, W be such subsets. Sice E 1 U, W BiE H U, W, ρ with mea E E 1 U, W c 2, by Lemma 2.1, l 3 l l 6 c 2 Pr E 1 U, W 2 l 3 l l 6 exp c 2 8 l 3 l l 6. Therefore, by uio boud we coclude that Pr U, W V H : U = W = s ad E 1 U, W 2 c 2 exp l 5 8 l 3 l l 6 2 exp l 4 c 2 8 l 3 = o1. l l 6 c 2 2 l 3 l l 6 To prove property iii, ote that E 1 U BiE H U, ρ with expectatio E E 1 U max{100 U l, 100 U 2 l 7 /}. Agai, by Lemma 2.2 ad uio boud we get that: { } Pr U V H : E 1 U max 1000 U l, 1000 U 2 l 7 / V H VH t t=1 t=1 exp exp t l 1000t l exp 999 l = o1. { } max 1000t l, 1000t 2 l 7 / For property iv, otice that E 1 Bi E H, ρ. By coditio 4, ad sice V H = Θ/ l 4, E H = O l K 8. So the expected size of E 1 is µ = O l K 8 ρ = o. Hece, agai, by Lemma 2.1 we coclude that E 1 = o with probability tedig to 1. 10

11 We have show that i the radomly chose subgraph the properties i iv hold a.a.s. I particular, there exists a istace where all hold. Now, we are ready to prove Theorem Proof [of Theorem 2.13] Let H 1 = V H, E 1 be a subgraph of H as give by Lemma For achievig his goal, Maker will play two games i parallel o E 1. I the odd moves Maker plays the 1 : 2b degree game o F 1 ad i the eve moves he plays as F-Breaker the 2b : 1 game E 1, F, where the wiig sets are F = { E 1 U, W : U, W V H, U W = ad U = W = l 5 l l 3 }. Combiig Claim 2.6 ad Lemma 2.14, Maker ca esure with his odd moves that Also, by Lemma 2.14 ii, for every v A 2 : d M H1 v = Ωl F /2b F F l Ω /l 3 l l 6 2 exp l 4 Ω l 3 = o1. l l 6 So by Theorem 2.3 Maker as F-Breaker wis the game E 1, F. disjoit subsets U, W V H of size U = W = V H l V H = Θ Maker ca claim a edge betwee U ad W. l 5 That is, for ay two = ω l 5 l l 3, Note that this gives coditio E2 of the expader defiitio, with R = V H / l V H. Furthermore, by Lemma 2.14 iv, the game lasts E 1 = o moves. To prove that by the ed of this game Maker s graph is ideed a V H / l V H, l l V H - expader, it remais to check coditio E1. Assume for a cotradictio that there exists a set X V H such that We distiguish three cases. X V H / l V H ad X N M X 2 l l V H X. 2.2 Case 1: X A 1 X /2. The X = O l 6 K by assumptio 2. Hece, ad by assumptio 4 ad 2.2, E H X, N M X E H X N M X { } max 100 X N M X l, 100 X N M X 2 p { } = O max X l l V H l, X l 2 l V H l 6. 11

12 But this implies E H X, N M X = o X l K 6 sice K > 12. However, sice every vertex v A 1 has Maker degree at least c 1 l K 6 we also coclude that E M X, N M X = Ωl K 6 X, a cotradictio. Case 2: X A 2 X /2 ad X <. By 2.1, for every v X A l 5 l l 3 2, d M H1 v = Ωl 2. Hece, E M H1 X, N M X = Ω X l 2. O the other had, by Lemma 2.14, { } E H1 X, N M X max 1000 X N M X l, 1000 X N M X 2 l 7 / { } = O max X l l l V H, X 2 l 2 l V H l 7 / = o X l 2, where the first equality follows from 2.2. But this, agai, is a cotradictio. Case 3: X V H l 5 l l 3 l V H. Sice Maker wis as F-Breaker the game E 1, F we coclude that N M X V H l 5 l l = Ω 3 l 4 = ωl l X, which cotradicts 2.2. This completes the proof. 3 The Perfect Matchig Game I this sectio we prove Theorem 1.1 ad a variat for radom bipartite graphs. Proof [of Theorem 1.1] First we describe a strategy for Maker ad the we prove it is a wiig strategy. At ay poit durig the game, if Maker caot follow the proposed strategy icludig the time limits the he forfeits the game. Before the game starts, Maker picks a subset U 0 V of size U 0 = l 4 such that for every v V, dv, U 0 = Ωl K 4. Such a subset exists because a radomly chose subset of size has this property by a l 4 Cheroff-type argumet a.a.s. Now, we divide Maker s strategy ito two mai stages. Stage I: At this stage, Maker builds a matchig M 0 of size /2 / l 4 which does ot touch U 0. Moreover, Maker wats to esure that by the ed of this stage, for every v V, d F v, U 0 = Ω l K 4, or d M v, U 0 = Ω l K Iitially, set M 0 =. For i, as log as M 0 < /2 / l 4, Maker plays his i-th move as follows: 1 If there exists a iteger j such that i = j l, the Maker plays the degree game 1 : b l DegV, U 0. 2 Otherwise, Maker claims a arbitrary free edge e i E s.t. e i e = for every e M 0 ad e i U 0 =. The, Maker updates M 0 to M 0 {e i }. 12

13 Whe Stage I is over, i.e. M 0 = /2 / l 4, Maker proceeds to Stage II. Stage II: Let V H = V \V M 0 such that V H = 2/ l 4, ad let H := G B[V H ]. We will show that H together with the subgraphs M cosistig of Maker s edges ad F cosistig of the free edges satisfies the coditios of Theorem That is, Maker ca play o H accordig to the strategy suggested by the theorem ad build a suitable expader i o moves. Ideed, stage I ad stage II costitute a wiig strategy, i.e. if Maker ca follow the proposed strategy, he will get a perfect matchig of G. By Theorem 2.13, Maker s subgraph of H will be a R, c-expader with R = V H / l V H ad c = l l V H, for large. By Theorem 2.12, this subgraph will be Hamilto-coected ad that is why it will cotai a perfect matchig M 1. Together with M 0 this forms a perfect matchig of G. Furthermore, Maker will wi i /2 + o moves, sice Stage I lasts at most /2 + o rouds, whereas i Stage II Maker eeds oly o moves. Thus, we oly eed to guaratee that Maker ca follow the strategy. By Lemma 2.10, the properties A1, A2 ad A3 hold a.a.s. for G. We coditio o these, ad heceforth assume that G satisfies A1, A2 ad A3, where f {1, l l 3 }. We cosider each stage separately. Stage I: First, cosider part 2, that is whe Maker tries to build the matchig M 0 greedily. Assume that Maker has to play his i-th move i Stage I ad i j l for ay j N. Furthermore, assume that still M 0 < /2 / l 4. Let T := V \ V M 0 U 0. The T > / l 4. Thus, by A3 f = 1, ET = ω. Sice i, Maker ad Breaker have claimed O edges so far. I particular, Maker ca fid a free edge i T to be added to M 0. Thus, he ca follow part 2 of Stage I. Secodly, cosider part 1. It is clear that Maker ca play the degree game. Thus, we oly eed to prove that the desired degree coditio 3.1 will hold. We already kow that there exists a costat c 1 > 0 with dv, U 0 c 1 l K 4 for every v V. If at the ed of Stage I Breaker has d B v, U 0 0.5c 1 l K 4 for some v V, the 3.1 holds trivially. Thus, we ca assume that d B v, U 0 0.5c 1 l K 4. I this case, Claim 2.6 gives d M v, U 0 0.1c 1 l K 6 /b. Stage II: We oly eed to check whether the coditios of Theorem 2.13 hold for H = V H, EH. Firstly, V H = 2/ l 4. Also, coditio 1 holds trivially by the defiitio of H. For coditio 2, ote that because of the degree coditio 3.1 we ca fid a costat c 2 such that V H = A 1 A 2, where A 1 := {v V H : d M v c 2 l K 6 } ad A 2 = {v V H : d F v c 2 l K 4 }. Sice stage I took at most rouds, A 1 = O l 6 K. Towards coditio 3, ote that by A3 f = l l 3 for every disjoit U, W V of l K 10 size, EU, W = Ω. Sice stage I took at most rouds, Breaker l 5 l l 3 l l 6 has claimed O edges. Hece, i the reduced graph where Breaker s edges are deleted, property 3 is satisfied. Coditio 4 follows by A2 ad sice H G. I the light of Theorem 1.3, i.e. the k-coectivity game, we would like to get a similar result for a radom bipartite graph. That is, for eve we deote by B,p a bipartite graph with two vertex classes of size /2, where every possible edge is iserted with probability p. We 13

14 show that Maker ca wi the perfect matchig game o B,p fast. The mai differece to the proof of Theorem 1.1 is that Maker will ot build a expader, but will rather fulfill the coditios of Lemma 2.9. Theorem 3.1 Let b 1 be a iteger, let K > 12, p = lk, ad let G B,p. The a.a.s. Maker wis the 1 : b perfect matchig game played o G withi 2 + o moves. Proof The proof is aalogous to the proof of Theorem 1.1, so we just sketch it here. For G = U 1 U 2, EG B,p, first choose a subset U 0 U 1 U 2 such that U 0 U 1 = U 0 U 2 = 2 l 4, ad dv, U 0 = Ωl K 4 for every v U 1 U 2. The Maker divides the game ito two stages. Stage I: Maker agai builds greedily a matchig M 0 of size /2 / l 4 which does ot touch U 0. Furthermore, Maker esures that by the ed of this stage for some c 1 > 0, d F v, U 0 c 1 l K 4 or d M v, U 0 c 1 l K 6 for every v U 1 U 2. Stage II: Let V H = V \ V M 0 with V H U i = l 4 ad let H = G B[V H]. Maker plays similarly to the strategy give i the proof of Theorem This time, he will ot build a expader like before. But he will esure that after o rouds his subgraph of H will satisfy coditios B1 ad B2 of Lemma 2.9 with r = V H / l V H. Similarly to Lemma 2.14, we fid a sparser subgraph H 1 H with the aalogue properties for bipartite graphs. As i the proof of 2.13, Maker plays i every eve move the 2b : 1 game E 1, F as F-Breaker where E 1 is the edge set of H 1, ad where F = { E 1 U, W : U U 1, W U 2 ad U = W = l 5 l l 3 }. Wiig this game, he will esure B2 with r = V H / l V H. To obtai B1, Maker plays i each odd move the 1 : 2b degree game. 4 The Hamiltoicity Game I this sectio we prove Theorem 1.2. Proof First we describe a strategy for Maker ad the we prove it is a wiig strategy. At ay poit durig the game, if Maker caot follow the proposed strategy icludig the time limits the he forfeits the game. As i the perfect matchig game, Maker picks a subset U 0 V of size U 0 = 10 l 4 such that for every v V, dv, U 0 = Ωl K 4. We divide Maker s strategy ito the followig four mai stages. Stage I: At this stage, Maker builds a matchig M 0 of size /2 /9 l 4 which does ot touch U 0. Moreover, Maker wats to esure that by the ed of this stage d F v, U 0 = Ω l K 4 or d M v, U 0 = Ω l K 6 for every v V. As soo as this stage is over, Maker proceeds to Stage II. Stage II: For a path P let EdP deote the set of its edpoits. Throughout this stage, Maker maitais a set M 1 of vertex disjoit paths, a subset M 2 M 1 ad a set Ed := {v 14

15 V : P M 1 \ M 2 such that v EdP }. Iitially, M 1 := M 0 ad M 2 =. Let r be the umber of rouds Stage I lasted. For every i > r, Maker will play his i-th move of this stage as follows: 1 If there exists a iteger j such that i = j l, the Maker plays the degree game 1 : b l DegV, U 0. 2 Otherwise, Maker claims a free edge xy betwee two vertices from Ed which are edpoits of two disjoit paths. Also, Maker updates M 1 by replacig the two old paths merged through xy by a ew oe. Note that this ew path is ot deleted from M 1. Maker also updates Ed accordigly. 3 If there is ay path P of legth at least 10 l K/3 the Maker updates M 2 := M 2 {P }. This stage eds whe M 1 = / l K/3. Thus, Stage II lasts ot more tha /2 + o rouds. Whe this stage eds, Maker proceeds to Stage III. Stage III: I this stage Maker esures that his graph o V \ U 0 will cotai a path P of legth at least / l 4. Moreover, Maker does so withi o moves. Let s be the umber of rouds Stage I ad II lasted. For every i > s, Maker plays his i-th move of this stage as follows: 1 If there exists a iteger j such that i = j l, the Maker plays the degree game 1 : b l DegV, U 0. 2 Otherwise, cosider the paths i M 1 of legth at least 3 l K/4. Maker tries to coect these paths, ot ecessarily through their edpoits, but through poits close to their eds. The full details of this partial game will be give i the proof below. Stage IV: Let x, y be the edpoits of P, the log path created i Stage III. Let V H = V \ V P {x, y}. At this stage Maker builds a Hamilto path o G B[V H ] with x, y as its edpoits. Moreover, Maker does so withi o moves. It is evidet that if Maker ca follow the proposed strategy the he wis the Hamiltoicity game withi + o moves. It thus remais to prove that ideed Maker ca follow the proposed strategy without forfeitig the game. By Lemma 2.10, the properties A1, A2 ad A3 hold a.a.s. for G. We coditio o these, ad heceforth assume that G satisfies A1, A2 ad A3, where f {1, l l 3 }. We cosider each stage separately. Stage I: The proof that Maker ca follow the proposed strategy for this stage is aalogous to the proof that Maker ca follow Stage I of the proposed strategy i the proof of Theorem 1.1. Stage II: Assume i j l. If M 1 > / l K/3, the M 1 \ M 2 = Ω / l K/3, sice there ca be at most /10 l K/3 disjoit paths of legth at least 10 l K/3. Hece, Ed = Ω / l K/3 ad by A3 f = 1, we have that the umber of edges of G spaed 15

16 by Ed is Ω l K/3 = ω. Therefore, we coclude that ideed Maker ca claim a free edge i G[Ed]. Stage III: Let U = {v V : v belogs to a path of legth 3 l K/4 i M 1 } ad update M 1 := M 1 \ {P : P is of legth 3 l K/4 }. Notice that U 3 l K/4 l K/3 = o l 4. So the sum of the legths of all paths i M 1 is at least V M 0 \ U /4 l For every path P M 1, defie LP ad RP to be the first ad last l K/4 vertices of P accordig to some fixed orietatio of the path. Notice that sice V P > 3 l K/4 it follows that LP RP = for every P M 1. Now, let m = / l K/2 ad let H = X, F be the hypergraph whose vertices are all edges of G B with both edpoits i P M 1 LP RP ad whose hyperedges are: { } m 2m F = E G\B S, T : distict P 1,..., P 2m M 1 s.t S = LP i, T = RP i. i=1 i=m+1 Note that for E G\B S, T F, S = T = m l K/4 = / l K/4 holds. Thus, by A3 f = 1, we have for a elemet of F that E G\B S, T E G S, T b = Ω l K/2. 2 Moreover, by A2, we get that X = O l K/4 M 1 p = O l 5K/6. Now, 2 Ω l 0.4K 2 F / l0.9k = F F F F M1 2 2 Ω l 0.4K m 2/ l e l K/6 K/2 2 Ω l 0.4K 2 exp l K/2 1 + K l l /6 Ω l 0.4K = o1. Thus, by Theorem 2.3 Maker as F-Breaker ca wi the l 0.9K, 1 game X, F. Lemma 2.4 therefore tells us that Maker ca claim at least oe elemet i every F F withi 1 + X /l 0.9K + 1 = o moves. To complete Stage III, let us defie the auxiliary directed graph D = V D, E D whose vertices are {P : P M 1 } ad whose directed edges are {P, Q : E M RP, LQ }. Notice that for every pair of disjoit subsets S, T V D such that S = T = m, there exists a edge i 16

17 D from S to T, sice Maker wis the game X, F. Now, we claim that Maker has a path of the desired legth i his graph. By Lemma 2.8, D cotais a directed path P = P 0... P t of legth t V D 2m + 1. Further, ote that ay path i M 1 has legth at most 20 l K/3. Combiig this with 4.1, removig paths P M 1 which do ot appear i P ad deletig uecessary parts of LP ad RP from paths P P we coclude that Maker has thus created a path of legth at least /4 l 4 2 M 1 l K/4 2m 20 l K/3 / l 4. Stage IV: Let P be the log path Maker has created i Stage III, ad let x, y be its edpoits. Deote V H = V \ V P {x, y}. Aalogously to the perfect matchig game, we ca use Theorem 2.13 ad Theorem 2.12 o H := G B[V H ]. That is, Maker ca build a expader o a sparse subgraph, ad thus obtais a Hamilto path i H with x, y as its edpoits i o moves. This completes the proof. 5 The k-coectivity Game I this sectio we prove Theorem 1.3. It is a simple applicatio of the Hamiltoicity game, the Perfect-matchig game o radom bipartite graphs, ad the degree game. Proof Let G G,p, ad radomly partitio the vertex set ito k disjoit sets V 1,..., V k 1, W where each V i has size k 1 W might be empty. For every 1 i k 1, let G i = G[V i ], ad for every 1 i < j k 1 let G ij be the bipartite subgraph of G with parts V i ad V j. From the defiitio it is clear that G i G k 1,p for every 1 i k 1 ad G ij B 2 k 1,p for every 1 i < j k 1. Now, Maker s strategy is to play the Hamiltoicity game o every G i, the perfect matchig game o every G ij, ad for every w W, Maker wats to claim k distict edges ww with w V \ W recall that G is typically such that dv = Θl K for every vertex v V G. Thus, i total Maker plays o t k 1 + k k 2 = k 2 + k k 2 boards. Eumerate all boards arbitrarily, ad let Maker play o board i mod t i his i-th move. Betwee ay two moves o a particular board, Breaker has claimed at most bk 2 ew edges o this board. Usig the trick of fake moves we ca assume that Maker plays the 1 : bk 2 Hamiltoicity game o every G i, the 1 : bk 2 perfect matchig game o every G ij, ad the degree-game 1 : bk 2 Deg{w}, V \ W for every w W. By Theorem 1.2 ad Theorem 3.1, every Hamiltoicity game ad every perfect matchig game lasts k 1 + o moves, whereas the games Deg{w}, V \ W last i total at most k W = O1 moves. If Maker succeeds o some board that is, either he formed a Hamilto cycle o some G j, or a perfect matchig o some G j1 j 2, or d M w, V \ W k for w W, the he quits playig o that particular board. That is, he igores this board ad plays o aother oe where he has ot wo yet. By Lemma 2.7 Maker is thus able to build a k-coected graph o G[V 1... V k 1 ]. Also, sice for every w W, δ M w, V \ W k, Maker s fial graph will be k-coected. I total, Maker plays at most k 1 + o + k 1 k o + O1 k k o moves, as claimed. 17

18 6 Ope problems We coclude with the list of several ope problems directly relevat to the results of this paper. Sparser graphs. For the three games cosidered i this paper, we would like to fid fast wiig strategies for Maker whe the games are played o G G,p, where p = 1+ε l for a costat ε > 0. Our proofs heavily deped o the ability of Maker to build a expader fast cf. Theorem 2.13, which does ot seem possible for such small p. We were ot able to prove a aalogue to Lemma 2.14 for smaller p s maily because of Property iv i this lemma. Therefore, we fid it very iterestig to either fid fast strategies for Maker substatially differet from ours, or alteratively provide Breaker with a strategy for delayig Maker s wi by a liear umber of moves. Faster wiig strategies for Maker. I this paper we have proved that Maker ca wi the perfect matchig game, the Hamiltoicity game ad the k-coectivity game played o G G,p withi /2 + o, + o ad k/2 + o moves, respectively. Although this is asymptotically tight it could be that the error term does ot deped o. It would be iterestig to fid the error term explicitly, or at least to provide tighter estimates o it. Fast wiig strategies for other games. It would be very iterestig to prove similar results, i.e. fast wiig strategies for Maker, for other games played o G G,p. We suggest the fixed-spaig-tree game. To be precise, let N be fixed, ad let T N be a sequece of trees o vertices with bouded maximum degree T. Maker s goal is to build a copy of T withi + o moves. Notice that this problem might be much harder tha what we proved sice eve the problem of embeddig spaig trees ito G G,p is still ot completely settled for more details see, e.g [15], [11]. Wiig strategies for Red. The problems cosidered i this paper were iitially motivated by fidig wiig strategies for Red i the strog games via fast wiig strategies for Maker see [8], [7]. It would be very iterestig to prove that ideed typically Red ca wi the aalogous strog games played o G G,p. Ackowledgemet: The authors wat to thak Peleg Michaeli for fruitful coversatios. Refereces [1] N. Alo ad J. H. Specer, The Probabilistic Method, Wiley, New-York, [2] J. Beck, Remarks o positioal games, Acta Math. Acad. Sci. Hugar , [3] J. Beck, Combiatorial Games: Tic-Tac-Toe Theory, Cambridge Uiversity Press, [4] I. Be-Eliezer, M. Krivelevich ad B. Sudakov, The size Ramsey umber of a directed path. Joural of Combiatorial Theory Series B, to appear. 18

19 [5] S. Be-Shimo, A. Ferber, D. Hefetz ad M. Krivelevich, Hittig time results for Maker-Breaker games. Proceedigs of the 22d Symposium o Discrete Algorithms SODA 11, [6] V. Chvátal ad P. Erdős, Biased positioal games, Aals of Discrete Math , [7] A. Ferber ad D. Hefetz, Wiig strog games through fast strategies for weak games, The Electroic Joural of Combiatorics, , P144. [8] A. Ferber ad D. Hefetz, Weak ad strog k-coectivity game. Preprit. Ca be foud at ferberas/papers. [9] A. Ferber, D. Hefetz ad M. Krivelevich, Fast embeddig of spaig trees i biased Maker-Breaker games, Europea Joural of Combiatorics , [10] D. Hefetz, M. Krivelevich ad T. Szabó, Hamilto cycles i highly coected ad expadig graphs, Combiatorica , [11] D. Hefetz, M. Krivelevich ad T. Szabó, Sharp threshold for the appearace of certai spaig trees i radom graphs, preprit. [12] D. Hefetz, M. Krivelevich, M. Stojakovic ad T. Szabó, Fast wiig strategies i Maker-Breaker games, Joural of Combiatorial Theory Series B, , [13] D. Hefetz ad S. Stich, O two problems regardig the Hamilto cycle game, The Electroic Joural of Combiatorics , R28. [14] S. Jaso, T. Luczak ad A. Ruciński, Radom graphs, Wiley, New York, [15] M. Krivelevich, Embeddig spaig trees i radom graphs. SIAM Joural o Discrete Mathematics , [16] A. Lehma, A solutio to the Shao switchig game, J. Soc. Idust. Appl. Math , [17] M. Stojaković ad T. Szabó, Positioal games o radom graphs, Radom Structures ad Algorithms , [18] D. B. West, Itroductio to Graph Theory, Pretice Hall,