Hitting time results for Maker-Breaker games

Transcription

1 Hittig time esults fo Make-Beake games Exteded Abstact Soy Be-Shimo Asaf Febe Da Hefetz Michael Kivelevich Abstact We aalyze classical Make-Beake games played o the edge set of a adomly geeated gaph G. We coside the adom gaph pocess ad aalyze, fo each of the popeties beig spaig k-vetex-coected, admittig a pefect matchig, ad beig Hamiltoia, the fist time whe Make stats havig a wiig stategy fo buildig a gaph possessig the taget popety (the so called hittig time. We pove that typically it happes pecisely at the time the adom gaph pocess fist eaches miimum degee k, ad 4, espectively, which is clealy optimal. The latte two statemets settle cojectues of Stojaković ad Szabó. We also coside a geeal-pupose game, the expade game, which is a mai igediet of ou poofs ad might be of a idepedet iteest. 1 Itoductio Let X be a fiite set ad let F X be a family of subsets. I the positioal game (X, F, two playes take tus i claimig oe peviously uclaimed elemet of X ad the game eds whe all of the elemets of X have bee claimed by eithe of the playes. The set X is ofte efeed to as the boad of the game. Positioal games have attacted a lot of attetio i the past decade ad a thoough itoductio to this field with a plethoa of esults ca be foud i a ecet moogaph of Beck []. I a Make-Beake-type positioal game, the two playes ae called Make ad Beake ad the membes of F ae efeed to as the wiig sets. Make wis the game if he occupies all elemets of some wiig set; The Blavatik School of Compute Sciece, Raymod ad Bevely Sackle Faculty of Exact Scieces, Tel Aviv Uivesity, 69978, Isael. soy@post.tau.ac.il. Reseach patially suppoted by a Faaju Foudatio Fellowship. School of Mathematical Scieces, Raymod ad Bevely Sackle Faculty of Exact Scieces, Tel Aviv Uivesity, 69978, Isael. febeas@post.tau.ac.il. Istitute of Theoetical Compute Sciece, ETH Züich, CH- 809 Switzelad. da.hefetz@if.ethz.ch. School of Mathematical Scieces, Raymod ad Bevely Sackle Faculty of Exact Scieces, Tel Aviv Uivesity, Tel Aviv 69978, Isael. kivelev@post.tau.ac.il. Reseach suppoted i pat by USA-Isael BSF gat 006, by gat 106/08 fom the Isael Sciece Foudatio, ad by a Pazy memoial awad. othewise Beake wis. We will always assume that Beake stats the game. We say that a game (X, F is a Make s wi if Make has a stategy (that ca be adaptive to Beake s moves that esues his wi i this game agaist ay stategy of Beake, othewise the game is a Beake s wi. Note that X ad F aloe detemie whethe the game is a Make s wi o a Beake s wi. A classical example of this Make- Beake settig is the popula boad game HEX. 1.1 Notatio Ou gaph-theoetic otatio is stadad ad follows that of [4]. I paticula, we use the followig. Fo a gaph G, let V (G ad E(G deote its sets of vetices ad edges espectively, ad let e(g = E(G. Fo a set A V (G, let E G (A deote the set of edges of G with both edpoits i A, ad let e G (A = E G (A. Fo disjoit sets A, B V (G, let E G (A, B deote the set of edges of G with oe edpoit i A ad the othe i B, ad let e G (A, B = E G (A, B. Fo a set S V (G, let N G (S = {u V (G \ S : v S, {u, v} E(G} deote the set of eighbos of S i V (G \ S. Fo a vetex w V (G, we abbeviate N G ({w} to N G (w. Fo a vetex w V (G \ S let d G (w, S = {u S : {u, w} E(G} deote the umbe of vetices of S that ae adjacet to w i G. We abbeviate d G (w, V \ {w} to d G (w which deotes the degee of w i G. The miimum vetex degee i G is deoted by δ(g. Fo a set S V (G let G[S] deote the subgaph of G with vetex set S ad edge set E G (S. Let c(g ad o(g espectively deote the umbe of coected compoets ad the umbe of coected compoets of odd cadiality i G. Lastly, we will deote by l(g the legth of a logest path i G, whee the legth of a path is the umbe of its edges. 1. Make-Beake games o gaphs Let G = (V, E be a gaph ad let P be a mootoe iceasig gaph popety o V (a family of gaphs o V, closed ude isomophism ad additio of edges. Coside the Make-Beake game (E, F P played o the edge set E as the boad of the game. The game is a wi fo Make if ad oly if the gaph spaed by the edges selected by Make thoughout the game satisfies the popety

2 P. We deote the family of gaphs G fo which the (E(G, F P game is a Make s wi by M P. Although the above game is descibed i game-theoetic tems, it should be oted that these games ae fiite pefect ifomatio games with o chace moves, ad M P is some gaph popety which clealy satisfies M P P. Moeove, sice P is mootoe iceasig, M P is clealy mootoe iceasig as well. By cosideig mootoe iceasig gaph popeties, the game ca be temiated as soo as the gaph spaed by Make s edges satisfies the popety, egadless of whethe all edges have bee claimed o ot. This leads to seveal atual questios. Fist, how spase ca a gaph G M P be? I this cotext, playig o adom gaphs (whee the desity of the gaph is cotolled by the distibutio o the gaphs becomes a vey atual questio. This settig was fomally iitiated i [] by Stojaković ad Szabó, ad this cuet wok is a futhe exploatio of it. Secod, oe ca also study the miimum umbe of moves eeded fo Make i ode to wi the game (see e.g. [, 1, 11, 1, 16], but wiig fast is ot i the focus of this cuet wok. 1. Radom gaphs The most widely used adom gaph model is the Biomial adom gaph, G(, p. I this model we stat with vetices, labeled, say, by V = {1,..., } = [], ad select a gaph o these vetices by goig ove all ( pais of vetices, decidig idepedetly with pobability p fo a pai to be a edge. The model G(, p is thus a pobability space of all labeled gaphs o the vetex set [] whee the pobability of such a gaph, G = ([], E, to be selected is p E (1 p ( E. This poduct pobability space povides us with a wide vaiety of pobabilistic tools fo aalyzig the behavio of vaious adom gaph popeties. (See moogaphs [6] ad [17] fo a thoough itoductio to the subject of adom gaphs. I the subsequet sectios we will eed at some poit to employ a slightly geealized model. Let F ( V be a abitay subset ad let G(, p F := G(, p \ F. Although the Biomial adom gaph model is vey atual ad elatively easy to use, it was ot the fist model to be cosideed. I thei semial pape, Edős ad Réyi cosideed the uifom pobability space ove all gaphs o a fixed set of vetices with exactly M edges, G(, M. Note that fo ay value of p, if we coditio the adom gaph G(, p to have exactly M edges, the we obtai exactly the Edős-Réyi adom gaph model. The similaity of the two models eables us to pove the occuece of evets i the G(, p model ad get the coespodig esult i the G(, M model. Popositio 1.1. ([17], Popositio 1.1 Let P = P( be a sequece of mootoe iceasig gaph popeties, 0 a 1 ad 0 M ( be a itege. If fo evey sequece p = p( [0, 1] such that p = M/ ( ± O (M (( ( M / it holds that lim P [G(, p P] = a, the lim P [G(, M P] = a. The covese esult to Popositio 1.1 holds 1 as well (see e.g. Popositio 1.1 i [17]; this eables us to tasfe esults fom oe model to the othe. Ufotuately, ot all popeties we will ecoute ad exploe ae mootoe iceasig, ad hece Popositio 1.1 caot be used i those cases. Noetheless, we would like to take advatage of the ease of calculatios i the G(, p model (due to the idepedece of appeaace of its edges, ad tasfe the esults to the G(, M model, fo the appopiate values of M. To achieve this we will use this moe cude estimate (see e.g. [17], which will suffice fo ou puposes. Popositio 1.. ([17], iequality (1.6 Let P be a( popety of gaphs o vetices ad let 1 M ( be a itege. Settig p = M/ we have P [G(, M P] M P [G(, p P]. Next, we coside the followig geeatio pocess of gaphs. Give a set V of vetices ad a odeig o the pais of vetices π : ( V [( ], we defie a gaph pocess to be a sequece of gaphs G = G(π = {G t } ( t=0 o V. Statig with G 0 = (V,, fo evey itege 1 t (, the gaph Gt is defied by G t := G t 1 π(t. Fo a give gaph pocess G o V, we defie the hittig time of a mootoe iceasig gaph popety P o V as τ( G; P = mi{t : G t P}. Whe selectig π uifomly at adom, the pocess G(π is usually called the adom gaph pocess. If G = {G t } ( t=0 evey 0 M is the adom gaph pocess, the, fo (, the gaph GM is distibuted accodig to G(, M, that is, G M G(, M. This etails that aalyzig the hittig time of a mootoe iceasig popety P is i fact a efiemet of the study of values of M ad p fo which G(, M P ad G(, p P espectively (whee to get the values of p we eed to use the covese of Popositio 1.1 as stated above. Fo evey positive itege k let δ k deote the gaph popety of havig miimum degee at least k, let EC k deote the gaph popety of beig k-edge coected, let VC k deote the gaph popety of beig k-vetex coected, ad let HAM deote the gaph popety of 1 I fact, whe movig fom G(, M to G(, p the mootoicity equiemet is ot ecessay

3 admittig a Hamilto cycle. Two coestoe esults i the theoy of adom gaphs ae that of Bollobás ad Thomaso [8] who poved that fo evey 1 k 1, with high pobability (o w.h.p. fo bevity τ( G; δ k = τ( G; EC k = τ( G; VC k, ad that of Komlós ad Szemeédi [18] who poved that w.h.p. τ( G; δ = τ( G; HAM (see also [5]. Note that these two esults (ad may othe which have succeeded povide a vey stog idicatio that the bottleeck fo such popeties i adom gaphs is i fact the vetices of miimum degee. The esults of this pape ae of the vey same atue. Befoe we poceed we stess that fo the sake of simplicity ad claity of pesetatio, we do ot make a paticula effot to optimize the costats obtaied i ou poofs. We also omit floo ad ceilig sigs wheeve these ae ot cucial. Most of ou esults ae asymptotic i atue ad wheeve ecessay we assume that is sufficietly lage. 1.4 Motivatio ad pevious esults Give a gaph G with miimum degee at most k 1 Beake ca keep claimig edges icidet to a vetex of miimum degee, ad with the advatage of playig fist will leave Make with a gaph cotaiig a vetex of degee at most k 1. This implies that Beake wis the k-edge-coectivity game (E(G, F ECk fo such gaphs, ad theefoe τ( G; M ECk τ( G; δ k fo evey gaph pocess G. I [] Stojaković ad Szabó wee the fist to coside Make-Beake games played o adom gaphs. By combiig theoems of Lehma [19] ad of Palme ad Spece [0], they obseved that fo evey fixed positive itege k, if G is the adom gaph pocess, the w.h.p. τ( G; M ECk = τ( G; δ k, thus povidig a vey pecise hittig time esult fo the edgecoectivity game. Similaly to the edge-coectivity case we have that fo evey gaph pocess G (1.1 τ( G; δ k τ( G; M VCk. Let PM deote the gaph popety of admittig a matchig of size / i a gaph o vetices. Evey gaph o G a eve umbe of vetices with miimum degee at most 1 is a wi fo Beake i the pefect matchig game (E(G, F PM. Hece, fo evey gaph pocess G o a eve umbe of vetices (1. τ( G; δ τ( G; M PM. I this pape, we say that a sequece of evets A i a adom gaph model occus w.h.p. if the pobability of A teds to 1 as the umbe of vetices teds to ifiity. I [] oly the case of k = 1 is explicitly metioed, but it ca be geealized fo ay positive itege k i a staightfowad mae. I [] Stojaković ad Szabó cojectued that if G is the adom gaph pocess, the w.h.p. equality holds i (1.. Although they did ot pove this cojectue, i 64 l [] they poved that if p >, the w.h.p. G(, p M PM. Note that this esult is optimal i p up to l +l l ω(1 multiplicative costat facto, fo if p, whee ω(1 is some fuctio which teds to ifiity with abitaily slowly, the w.h.p. δ(g(, p 1, ad hece by (1., w.h.p. G(, p / M PM. Clealy, evey gaph G with miimum degee at most is a wi fo Beake i the Hamiltoicity game (E(G, F HAM. Hece, we have that fo evey gaph pocess G (1. τ( G; δ 4 τ( G; M HAM. I [] Stojaković ad Szabó cojectued that if G is the adom gaph pocess, the w.h.p. equality holds i (1.. Oe of the fist esults i the field of Make-Beake games o gaphs is due to Chvátal ad Edős i thei semial pape [9], which states that K M HAM fo sufficietly lage values of (i [16] the thid autho ad Stich poved that 8 suffices. The poblem of fidig spase gaphs which ae a wi fo Make was addessed by Hefetz et. al. [15] whee they showed that, fo sufficietly lage values of, thee exists a gaph G M HAM o vetices with e(g 1. Playig the Hamiltoicity game (E(G, F HAM o the adom gaph G(, p was fist cosideed i the oigial pape of Stojaković ad Szabó [] whee they poved that l if p >, the w.h.p. G(, p M HAM. Late, Stojaković [] foud the coect ode of magitude povig that p > 5.4 l / suffices fo G(, p to be w.h.p. Make s wi i the Hamiltoicity game. This equiemet o p was subsequetly impoved to p l +(l l s, whee s is some lage but fixed costat, by Hefetz et. al. [14]. Note that this esult is vey l + l l ω(1 close to beig optimal, fo if p =, whee ω(1 is some fuctio which teds to ifiity with abitaily slowly, the w.h.p. δ(g(, p < 4 ad hece by (1. w.h.p. G(, p / M HAM. Lastly, i [4] the fist ad fouth authos with Sudakov studied the Hamiltoicity game played o the edges of adom egula gaphs (the uifom pobability measue ove all d-egula gaphs o a fixed vetex set ad poved that fo lage eough costat values of d this game is Make s wi. 1.5 Ou esults I this pape we addess the above metioed Make-Beake games o adom gaphs, amely whe Make s goal is to build gaphs which satisfy the popeties of beig k-vetex coected, admittig a pefect matchig, ad beig Hamiltoia. Specif-

4 ically, the mai objective of this pape is to pove that the tivial miimum degee equiemet as stated i (1.1, (1., ad (1. is actually the bottleeck fo a typical adom gaph to be a wi fo Make i all of the above metioed games. The followig esults will thus be poved. Theoem 1.1. Fo evey fixed itege k 1, if G is the adom gaph pocess, the w.h.p. τ( G; M VCk = τ( G; δ k. Fo evey positive itege k it holds that VC k EC k, hece Theoem 1.1 is i fact a impovemet of the afoemetioed esult of Stojaković ad Szabó i []. We also ote that, by usig the theoem of Lehma [19], we ca get the esult of Palme ad Spece [0] fo eve values of k as a coollay of Theoem 1.1. The followig esult fo the pefect matchig game is also poved. Theoem 1.. If G is the adom gaph pocess o a eve umbe of vetices, the w.h.p. τ( G; M PM = τ( G; δ. Theoem 1. settles a cojectue aised i []. By the coectio betwee the adom gaph models as descibed i Sectio 1. ad by kow esults o the distibutio of the miimum degee of G(, p, Theoem 1. implies that w.h.p. G(, p M PM fo evey l +l l +ω(1 p, whee ω(1 teds abitaily slowly to ifiity with, impovig o the esult of Stojaković ad Szabó i []. Theoem 1.. If G is the adom gaph pocess, the w.h.p. τ( G; M HAM = τ( G; δ 4. Theoem 1. settles a cojectue aised i []. Moeove, similaly to the above, Theoem 1. impoves o the esult of Hefetz et. al. i [14] by implyig that l + l l +ω(1, w.h.p. G(, p M HAM fo evey p whee ω(1 teds abitaily slowly to ifiity with. We ote that, by usig a slight modificatio of ou poofs, Theoems 1. ad 1. ca i fact be exteded. Fo evey positive itege k 1, let PM k ad HAM k deote the gaph popeties of admittig k paiwise edgedisjoit pefect matchigs, ad k paiwise edge-disjoit Hamilto cycles. Theoem 1.4. Fo evey fixed itege k 1, if G is the adom gaph pocess, the w.h.p. τ( G; M PM k = τ( G; δ k. Theoem 1.5. Fo evey fixed itege k 1, if G is the adom gaph pocess, the w.h.p. τ( G; M HAM k = τ( G; δ 4k. Theoem 1.5 ca be viewed as a Combiatoial game aalog of the classical esult of Bollobás ad Fieze [7] who poved that w.h.p. τ( G; HAM k = τ( G; δ k (see also [1] fo a extesio to o-costat miimum degee i the G(, p model. 1.6 Ogaizatio The est of the pape is ogaized as follows. Sectio is devoted to the itoductio of expade gaphs ad the aalysis of a geeal game i which Make s goal is to build such a gaph. This will give us a famewok fom which we ca build o to pove the cocete esults o the moe atual games metioed above. We the move o to povide the full poofs of Theoems 1.1 ad 1. ad 1. i Sectio. These poofs will ely heavily o the geeal expade game ad the popeties of adom gaphs ad adom gaph pocesses which we discussed i the pecedig two sectios. Sectio 4 pesets a sketch of a poof of Theoems 1.4 ad 1.5 as these poofs follow quite closely the footsteps of the poofs peseted i Sectio. I the Appedix we povide some additioal backgoud ad techical details which will be of use thoughout the couse of ou poofs. A expade game o pseudo-adom gaphs ad its applicatio to adom gaphs The mai object of this sectio is to descibe a geeal Make-Beake game which will eside i the coe of all of ou poofs. Let us fist defie the type of expades we wish to study. Defiitio.1. Fo evey c > 0 ad evey positive itege R we say that a gaph G = (V, E is a (R, c- expade if evey subset of vetices U V of cadiality U R satisfies N G (U c U. We deote the gaph popety of beig a (R, c-expade by X R,c. Remak.1. Fom the above defiitio it clealy follows that fo evey c > 0 ad evey positive itege R (both c ad R ca be fuctios of the umbe of vetices of the gaph i questio, the gaph popety X R,c is mootoe iceasig. Next, we coside some stuctual popeties of (R, c-expades. The followig two popositios show that the emoval o additio of subsets that satisfy cetai popeties esult i gaphs that ae still expades. These popeties will allow us to slightly modify cetai expades without losig thei expasio popeties. Popositio.1. If G = (V, E is a (R, c-expade ad U V is a subset of vetices such that o two vetices of U have a commo eighbo i G, the G[V \ U] is a (R, c 1-expade.

5 Poof. Let S V \ U be a set of cadiality S R. It follows by ou assumptio o U that N G (v U 1 holds fo evey vetex v S. Hece N G[V \U] (S N G (S S (c 1 S. Popositio.. Let G = (V, E be a gaph, let c > 0, ad let R be a positive itege. Let U V be a subset of vetices such that N G (U (c 1 U fo evey U U, ad, moeove, thee is o path of legth at most 4 i G whose (possibly idetical edpoits lie i U. If G[V \ U] is a (R, c-expade, the G is a (R, c 1-expade. Poof. Let V = V \ U ad let H = G[V ]. Let S V be of cadiality s R, ad set S 1 = S U ad let S = S \ S 1 with espective cadialities s 1 ad s = s s 1. Ou assumptio o U implies, i paticula, that it is idepedet. It follows that N G (S 1 V \ U. Futhemoe, N G (S 1 ca cotai at most oe vetex fom each set {{t} N H (t} t V, ad hece N G (S 1 (S N H (S S. It follows that N G (S N G (S 1 (N H (S \(N G (S 1 (S N H (S, ad that N G (S (c 1s 1 + (c s s = (c 1s 1 + (c 1(s s 1 = (c 1s as claimed. Next, we descibe some sufficiet coditios fo a gaph G = (V, E to be a expade (with appopiate paametes. M1 e G (U δ(g U (c+1 fo evey subset of vetices U V of cadiality 1 U < (c + 1; M e G (U, W > 0 fo evey pai of disjoit subsets of vetices U, W V of cadiality U = W =. Lemma.1. Fo evey c > 0, if G = (V, E is a gaph which satisfies popeties M1 ad M fo some positive itege V V c+, the G is a ( c+1, c-expade. V Poof. Set R = c+1 ; ote that R holds by the assumptio of the lemma. Assume fo the sake of cotadictio that thee exists a set S V of cadiality S R fo which N G (S < c S. Let T = S N G (S, the T < (c+1 S. If 1 S, the T < (c+1. Moeove, sice all edges that have at least oe edpoit i S ae spaed by the vetices of T, it follows that e G (T δ(g S > δ(g T (c+1, which cotadicts popety M1. If < S R, the, sice e G (S, V \ T = 0 ad V \ T > V (c + 1 S V (c + 1R =, we obtai a cotadictio to popety M. This cocludes the poof of the lemma. The easo we study (R, c-expades is the fact that they etail some pseudo-adom popeties fom which (ude some coditios o R ad c all of the atual popeties that ae cosideed i this pape, amely, admittig a pefect matchig, beig k-vetexcoected ad beig Hamiltoia, follow. We will povide a sufficiet coditios fo a (R, c-expade to be k-vetex coected ad to admit a pefect matchig. Hece by playig fo a (R, c-expade, Make will be able to wi the two games whose goals ae the afoemetioed two popeties (each posig diffeet coditios o R ad c. The sufficiet coditio fo a gaph to be Hamiltoia, that we will use i the couse of the poof, is moe delicate tha the coditios fo k-vetex coectivity ad fo admittig a pefect matchig, ad equies some additioal ideas, but the cux of the poof will still ely o expades. Next, we povide sufficiet coditios fo G M XR,c, o amely, fo a gaph G to be Make s wi whe Make s goal is to build a (R, c-expade. Although this game may seem at fist to be a uatual ad atificial game to study, it tus out that this game will lie i the heat of ou poofs of all of the esults peseted i this pape. Give paametes c > 0, 0 < ε < 1, K > 0 ad a positive itege V c+1, we defie the followig two popeties of a gaph H = (V, E o vetices. These popeties, which ae closely elated to popeties M1 ad M, will be eeded i the poof of the mai esult of this sectio. Q1 e H (U εδ(h U 10(c+1 fo evey subset of vetices U V of cadiality 1 U < (c + 1; ( Q e H (U, W K l fo evey pai of disjoit subsets of vetices U, W V of cadiality U = W =. Remak.. Wheeve we will cite popety Q we will give a explicit expessio fo K which will ot ecessaily be a costat. Theoem.1. Thee exists a itege 0 > 0 such that fo evey gaph G = (V, E o 0 vetices with miimum degee δ(g > 0 ad fo evey choice 1 of paametes δ(g < ε < 1, c > 0, ad itege 0 < mi{ c+, e } fo which G satisfies popeties 0 Q1 ad Q with K = (1 ε, Make ca wi the ( c+1, c-expade game o G, that is, G M XR,c with R = c+1. The mai idea of the poof is to show that we ca split the gaph G ito two pats whee oe has a lage eough miimal degee ad the othe satisfies Q fo some othe value of K. Fom the miimal degee of the fist gaph ad the Q1 popety of G

6 we deive the popety M1. To guaatee the popety M we esot to a classical esult i Make-Beake theoy, amely the Edős-Selfidge citeia which we itoduce i Sectio A of the Appedix as Theoem A.. Ou poof of Theoem.1 will be peseted as a seies of thee lemmata (Lemmata.,., ad.4 whose compositio implies the theoem diectly. Lemma.. Thee exists a itege 0 > 0 such that fo evey gaph G = (V, E o 0 vetices with miimum degee δ(g > 0 ad fo evey choice of 1 paametes δ(g < ε < 1 ad itege 0 < /e 4 fo which G satisfies popety Q with K = (1 ε, the edge set E ca be split ito two disjoit subsets E = E 1 E such that the gaph G 1 = (V, E 1 has miimum degee δ(g 1 εδ(g ad the gaph G = (V, E satisfies popety Q with K =. Poof. Pick evey edge of G to be a edge i G 1 with pobability ε idepedetly of all othe choices. The degee i G 1 of evey vetex v V is biomially distibuted, that is, d G1 (v Bi(d G (v, ε ad thus its media is at least εδ(g. By ou choice of ε we have that εδ(g > εδ(g ad theefoe P [d G1 (v εδ(g ] > 1/. Sice the degees of evey two vetices ae positively coelated, we have that P [δ(g 1 εδ(g ] >. Let U, W be a pai of disjoit subsets of vetices of cadiality U = W =. By ou assumptio o G we have that e G (U, W l ( 1 ε. As e G (U, W Bi(e ( G (U, W, 1 ε we have E [e G (U, W ] l. Applyig Theoem A.1 we have [ ( ] P e G (U, W < l ( ( 1 l exp exp ( l. By applyig the uio boud ove all pais of disjoit subsets of vetices of cadiality each, we coclude that the pobability that G violates popety Q with K = is at most ( ( exp ( l ( e exp ( ( = exp 1 + l ( l exp <, 4 ( l l ( ad theefoe thee exists a patitio of G as claimed. The followig lemma povides a sufficiet coditio o a gaph G = (V, E fo it to be a Make s wi i the game (E, F M, that is, the game o G i which Make s goal is to build a subgaph which satisfies the (mootoe iceasig popety M. I ode to pove this esult, we ivoke a athe stadad techique of studyig a dual game i which the oles of Make ad Beake ae exchaged. Note that i the dual game, Beake (which was the oigial Make is the secod playe. Lemma.. Thee exists a itege 0 > 0 such that fo evey gaph G = (V, E o 0 vetices ad fo evey itege 0 < /e 0 fo which G satisfies popety Q with K =, playig o E Make ca build a subgaph of G which satisfies popety M. Poof. Let G be ay gaph with vetex set V. I ode fo Make to build a gaph which satisfies popety M, he ca adopt the ole of Beake i the game (E, L, whee L is the family of edge-sets of all iduced bipatite subgaphs of G with both pats of size. Recall that, by popety Q with K =, evey such wiig set ( L L spas at least l L L L U V ; U = W V \U ; W = ( ( ( exp l ( e ( ( exp l ( ( ( exp l < 1. edges. It follows that e G (U,W ( l + l l l ( The assetio of the lemma follows eadily by Theoem A..

7 Lemma.4. Thee exists a itege 0 > 0 such that fo evey gaph G = (V, E o 0 vetices ad fo evey choice of paametes 0 < ε < 1, c > 0 ad itege 0 < c+ fo which G satisfies popety Q1 ad whose edge set ca be patitioed ito two disjoit sets E = E 1 E whee G 1 = (V, E 1 is of miimum degee δ(g 1 ε δ(g, ad G = (V, E satisfies Q with K =, Make ca wi the ( c+1, c-expade game, that is, G M XR,c with R = c+1. Poof. Befoe the game stats, Make splits the boad ito two pats, G 1 = (V, E 1 ad G = (V, E as idicated i the lemma. Make the plays two sepaate games i paallel, oe o E 1 ad the othe o E. I evey tu i which Beake claims some edge of E i, fo i = 1,, Make espods by claimig a edge of E i as well (except fo maybe oce if Beake has claimed the last edge of E i. Let H deote the gaph built by Make by the ed of the game ad set H 1 = (V, E(H E 1 ad H = (V, E(H E. The game o E 1 is played accodig to Lemma A.. Hece, at the ed of the game, Make s gaph H 1 will have miimum degee at least δ(h 1 δ(g1 5. Sice G satisfies popety Q1 ad δ(g 1 εδ(g it follows that, fo evey U V of cadiality 1 U < (c + 1, the umbe of Make s edges with both edpoits i U is e M (U e G (U εδ(g U δ(h U (c+1 10(c+1 δ(g1 U 10(c+1 δ(h1 U (c+1. Hece, H satisfies popety M1. The game o E is played accodig to Lemma., ad theefoe at the ed of the game, Make will build a gaph H which satisfies popety M. By the mootoicity of M, this popety also holds fo H. Notig that H,, ad c satisfy the coditios of Lemma.1, we deduce that H M XR,c, that is, Make s gaph is a (R, c-expade as claimed. Let G = (V, E be a gaph o vetices ad fo a positive itege t deote D t (G = {v V : d G (v < t}. Note that if G is a subgaph of H the D t (G D t (H. Based o Theoem.1 we show that emovig vetices of small degee fom the adom gaph typically leaves a gaph o which the expade game ca be wo by Make. I fact, we eve show that Make ca wi the game whe this gaph is thied substatially (i.e. the vast majoity of the edges ae emoved. Note that this stoge popety (i.e. of the existece of a good thiig of the gaph will play a cucial ole i the poof of Theoem 1.. To this ed we will eed some stuctual popeties of the set D t i the adom gaph model G(, M, which we have see to be equivalet to stoppig the adom gaph pocess at G M. The followig popositios ae cetal i all of ou agumets, as we will eed to take special cae of vetices of small degee. Popositio.. Fo evey itege t l 0.9, if G = {G i } ( i=0 is the adom gaph pocess ad M τ( G; δ 1 the w.h.p. D t (G M 0.. The poof of this popositio will appea i the full vesio of this pape. Popositio.4. Fo evey fixed iteges k 1 ad t l 0.9, if G = {G i } ( i=0 is the adom gaph pocess ad M = τ( G; δ k the w.h.p G = G M does ot cotai a path of at most 4 distict edges whee both (possibly idetical edpoits lie i D t (G M. I ode to pove the above esults ad othe stuctual popeties of the adom gaph pocess we esot to the use of G(, p, whee the aalysis is much simple, ad use Popositio 1. to tasfe the esults to the oigial adom gaph model. The full poof of Popositio.4 will appea i the full vesio of this pape, but we povide hee a bief sketch of it. Fist oe ca use Popositio. to show that w.h.p. the set U = D t (G mk is small. At time m k, oe ca also pove that thee is o udesied shot path ad that w.h.p. at time M k all degees ae O(log (usig, say, Popositio 1.. Note that it follows that whe uig the adom gaph pocess fom time m k to M k, while assumig that the fial poit satisfies the above coditio, we have that duig the u all degees ae still O(log. Now, coside the pobability that the ith edge, e i, of the pocess, fo ay m k < i M k closes a udesied path (a path of legth at most 4 i G i betwee two vetices of U. Fo this to happe both edpoits of e i should be at distace at most i G i fom the set U. Sice U is small, ad all degees ae O(log i G i, the umbe of such pais is O(( U log = o( 0.8, ad theefoe the pobability that e i is such is at most o( 0.8 / < 1.. The oe ca apply a uio boud agumet ove all steps i betwee m k ad M k to complete the poof. The followig lemma, whose poof will appea i the full vesio of this pape, guaatees that afte the emoval of vetices of small degee, Make ca wi the expade game whe R ad c ae withi a specified age. The mai idea is to show that oce the vetices of small degee ae emoved fom the gaph, all of the coditios of Theoem.1 ae met with high pobability. This lemma is the mai stuctual popety of the adom gaph pocess which we use i the couse of the poof of Theoem 1.. Lemma.5. Fo evey α > 0 ad fixed positive itege k if G = {Gi } ( i=0 is the adom gaph pocess

8 ad M = τ( G; δ k the w.h.p. G = (V, E = G M \ D l 0.9 (G M o vetices cotais a spaig subgaph Ĝ G M with at most l 0.97 edges such that Ĝ M XR,c fo evey c l 0.0 ad R (1 α c+1. Remak.. As was oted i Remak.1, by the mootoicity of X R,c, the above Lemma ca be used to deduce that the gaph G M XR,c. Poofs of Theoems 1.1, 1. ad 1. This sectio is devoted to the poofs of Theoems 1.1, 1. ad 1.. These thee theoems make use of some simple sufficiet coditio o (R, c-expades to have the equied popety at had ad some simple applicatios of the popeties of adom gaphs as peseted i the pevious sectios. Lemma.1. Fo evey positive itege k, if G = (V, E is a (R, c-expade such that c k, ad Rc 1 ( V + k, the G VC k. Poof. Assume fo the sake of cotadictio that thee exists some set S V of size k 1 whose emoval discoects G. Deote the coected compoets of G \ S by S 1,..., S t, whee t ad 1 S 1... S t. If S 1 R, the k 1 = S N G (S 1 c S 1 c k, which is clealy a cotadictio. Assume the that S 1 > R. Fo i {1, }, let A i S i be a abitay subset of size R. It follows that V S 1 S N G (S 1 N G (S N G (A 1 N G (A = N G (A 1 + N G (A N G (A 1 N G (A Rc S V + 1, which is clealy a cotadictio. It follows that G is k-vetex-coected as claimed. I ode to pove Theoem 1.1 it thus suffices to show that w.h.p. at the momet the adom gaph pocess fist eaches miimum degee k, Make has a wiig stategy fo the (R, c-expade game fo suitably chose values of R ad c. I doig so we will heavily ely o Theoem.1. Poof. [Poof of Theoem 1.1] Fix some positive itege k 1 ad let G = {G i } ( i=0 deote the adom gaph pocess. Set M = τ( G; δ k, let G = G M, Small = D l 0.9 (G, G = G[V \ Small] ad deote by the umbe of vetices i G. Settig c = k +, ad R = k+4, the coditios of Lemma.5 ae met, ad thus G M X. k+4,k+ Make s stategy is quite atual. He splits the boad ito F 1 = E(G ad F = E G (Small, V \ Small, ad plays the coespodig two games i paallel, that is, i each move Make will claim a edge of the boad Beake chose his last edge fom (except fo possibly his last move i oe of the two games. Playig o the edges of F 1, Make aims to build a ( k+4, k + -expade. As oted above, Make has a wiig stategy fo this game. Playig o the edges of F, Make follows a simple paiig stategy which guaatees that, by the ed of the game, the gaph H which Make costucts will satisfy d H (v d G (v/ fo evey v Small. To achieve this goal, wheeve Beake claims a edge which is icidet with some vetex v Small, Make espods by claimig a diffeet edge icidet with v if such a edge exists, ad othewise he claims a abitay fee edge of F 1 F. Sice the miimum degee i G is k, it follows by Make s stategy fo the game o F ad by Popositio.4, that i Make s gaph H, the vetices of Small fom a idepedet set with k edges emittig out of each vetex. Sice the gaph H = H[V \ Small] is a ( k+4, k +-expade, ad sice (k + k+4 1 (+k holds fo evey k 1 by Popositio., Lemma.1 implies that H VC k. Addig to H the vetices of Small with thei icidet edges clealy keeps the k- vetex coectivity popety, as coectig a ew vetex to at least k vetices of a k-vetex coected gaph poduces a k-vetex coected gaph. This cocludes the poof of the theoem. I ode to show that expasio etails admittig a pefect matchig, we make use of the well-kow Bege- Tutte fomula fo the size of a maximum matchig i a gaph (see e.g. [4, Coollay..7]. Theoem.1. (Bege-Tutte The maximum umbe of vetices which ae satuated by a matchig i a gaph G = (V, E is mi S V { V + S o(g S}. The followig lemma is applicable egadless of the paity of the umbe of vetices i the gaph. Lemma.. If G = (V, E is a (R, c-expade such{ that c ad }(c + 1R V mi R c(c 1 c+r(c 1, Rc(c 1 6c c 1, the G PM. Poof. Fom the coditios o R ad c it follows that Rc > V / ad, combied with G beig a (R, c- expade, this tivially implies that the gaph G must be coected. Settig S =, we have that o(g S = 1 fo odd V, ad that o(g S = 0 fo eve V. By Theoem.1 we ca thus assume that S. We will i fact pove that S c(g S holds fo evey o-empty S V. It clealy suffices to pove this fo evey S V of cadiality S V /. Let S be such a set, let t = c(g S, ad let S 1,..., S t deote the coected compoets of G S, whee

9 1 S 1... S t. Assume fist that thee exists a set A {1,..., t} such that S /c < i A S i R. By defiitio we have N G ( i A S i S. It follows that S N G ( i A S i c i A S i > S, which is clealy a cotadictio. Hece, o such A {1,..., t} exists. It follows that thee must exist some 0 j t such that j i=1 S i S /c ad S i > R S /c fo evey j < i t. If j t 1, the, sice S i 1 fo evey 1 i t, it follows that t t 1 i=1 S i + 1 S /c + 1 S. Hece, we ca assume that j t. We claim that, ude this assumptio, S Rc. Ideed, assume fo the sake of cotadictio that 1 S < Rc o equivaletly, that c(r S /c > S. If R S /c S j +1 R, the, as S N G (S j +1 we have that S N G (S j +1 c(r S /c > S, a cotadictio. Theeofoe, S i > R fo j < i t. Sice j t, fo i {t 1, t}, we ca choose A i S i to be a abitay subset of size R. It follows that V S t 1 S t N G (S t 1 N G (S t N G (A t 1 N G (A t = N G (A t 1 + N G (A t N G (A t 1 N G (A t Rc S > V, which is, agai, clealy a cotadictio. We deduce that V /4 < Rc/ S V / < Rc. Note that ude ou assumptio o R ad c we have that R S /c > 4, ad theefoe S i 5 fo all j < i t. Moeove, as all S i cotai at least oe vetex j S /c. Puttig this togethe we have that S > 1 t i=1 S i j +(t j (R S /c 5t 4j, ad theefoe t < S 5 (+ 4 c S which completes the poof of the lemma. I ode to pove Theoem 1. we poceed vey similaly to the poof of Theoem 1.1. Poof. [Poof of Theoem 1.] Let G = {G i } ( i=0 deote the adom gaph pocess. Set M = τ( G; δ, let G = G M, Small = D l 0.9 (G, G = G[V \ Small] ad deote by the umbe of vetices i G. Settig c = 8, ad R = 10, the coditios of Lemma.5 ae met, ad thus G M X. 10,8 Make s stategy follows the same lies as that of the k-coectivity case peseted i the poof of Theoem 1.1. He splits the boad ito F 1 = E(G ad F = E G (Small, V \ Small, ad plays the coespodig two games i paallel, that is, i each move Make will claim a edge of the boad Beake chose his last edge fom (except fo possibly his last move i oe of the two games. Playig o the edges of F 1, Make aims to build a ( /10, 8-expade. As oted above, Make has a wiig stategy fo this game. We deote the estictio of the gaph built by Make by the ed of the game to the edges of F 1 by H 1. Playig o the edges of F, Make follows a simple paiig stategy which guaatees that, by the ed of the game, the gaph H which Make costucts will satisfy d H (v d G (v/ fo evey v Small. To achieve this goal, wheeve Beake claims a edge which is icidet with some vetex v Small, Make espods by claimig a diffeet edge icidet with v if such a edge exists, ad othewise he claims a abitay fee edge of F 1 F. Recallig Popositio.4 we ca assume that Small is a idepedet set i G ad that o two vetices i Small shae a commo eighbo. As the miimum degee i G is, Make s gaph, H = H 1 H, will cotai at least oe edge emittig out of evey vetex i Small, each icidet with a diffeet vetex of V \ Small. Theefoe, thee exists a matchig i M which coves all vetices of Small. Let T deote the set of vetices of V \ Small which ae coveed by M. Agai, by Popositio.4 we ca assume that o two vetices i T shae a commo eighbo (as this would ceate a path of legth 4 betwee two vetices i Small. Sice, the gaph H is a ( /10, 8-expade, it follows by Popositio.1 that the gaph H = H \T is a ( /10, 7-expade. The values R = /10 ad c = 7 satisfy the coditio of Lemma., implyig that H PM. Let M be some pefect matchig of H, the M M is a pefect matchig of H. This cocludes the poof of the theoem. Ou poof of Theoem 1. is faily simila to the two the poofs of Theoems 1.1 ad 1.. Howeve, afte havig built a appopiate expade, Make will eed to claim additioal edges i ode to tasfom his expade ito a Hamiltoia gaph. I ode to descibe the elevat coectio betwee Hamiltoicity ad (R, c-expades, we equie the otio of boostes. Defiitio.1. Fo evey gaph G we say that a oedge {u, v} / E(G is a booste with espect to G if G + {u, v} is Hamiltoia o l(g + {u, v} > l(g. We deote by B G the set of boostes with espect to G. The followig is a well-kow popety of (R, - expades (see e.g. [1]. Lemma.. If G is a coected o-hamiltoia (R, -expade the B G R /. Ou goal is to show that duig a game o a appopiate gaph G, assumig Make ca build a subgaph of G which is a (R, c-expade, he ca also claim sufficietly may such boostes, so that his (R, c- expade becomes Hamiltoia. I ode to do so, we futhe aalyze the stuctue of the adom gaph pocess. Lemma.4. If G = {Gi } ( i=0 is the adom gaph pocess ad M = τ( G; δ 4, the w.h.p. G M does ot

10 cotai a coected o-hamiltoia (/5, -expade Γ with at most l 0.98 edges such that E(G M B Γ l Poof. Fist we ote that ay (/5, -expade must be coected, as each coected compoet must be of size at least /5 + /5 > /. Let m 4 M M 4 be a itege, let p = M / ( > l, ad let G = (V, E G(, p. Ou goal is to pove that the pobability that G cotais a coected o-hamiltoia (/5, - expade subgaph Γ with at most l 0.98 edges such that E B Γ l 0.98 is much smalle tha the pobability that e(g = M. Summig ove all itegal values of M i the iteval [m 4, M 4 ], ad applyig Popositio 1. to each of these values, will eable us to complete the poof. Let S deote the set of all labeled o-hamiltoia (/5, -expades o the vetex set V which have at most l 0.98 edges. Fix a gaph Γ = (V, F S, the clealy P [Γ G] = p F. Now, let G = (V, E \ F G(, p F. By defiitio, evey booste with espect to Γ is a o-edge i Γ, hece B Γ is a subset of the potetial pais of the gaph G. Lemma. implies that B Γ /50, ad sice E(G B Γ Bi( B Γ, p, it follows that E [ E(G B Γ ] p Theoem A.1 we have 50 > l P [ E(G B Γ l 0.98 ] ( ( 1 50 l exp p ( exp p Applyig Next, we ote that by the idepedece of appeaace of edges i G(, p, the evet Γ G ad the evet that some booste e with espect to Γ was chose amog the edges of G, ae idepedet evets. We ca thus use a uio boud agumet by goig ove all Γ S to uppe boud the pobability that G cotais a coected o- Hamiltoia (/5, -expade Γ with at most l 0.98 edges, such that E B Γ l 0.98 as follows l 0.98 m=1 l 0.98 m=1 l 0.98 m=1 exp (( m ( p m exp p 101 ( e m ( p exp p m 101 ( ( exp m 1 + l ( p. 10 ( p p m 101 Usig Popositio 1., the above calculatio implies that the same evet, with G G(, M, is uppe bouded by ( M exp p 10 exp ( l 10. Takig the uio boud ove all itegal values of m 4 M M 4, we coclude that the pobability thee exists such a itege M fo which G M violates the claim is at most (M 4 m exp ( l 10 l l l exp ( l 10 = o(1. We ae ow eady to peset the full poof of Theoem 1.. Poof. [Poof of Theoem 1.] Let G = {G i } ( i=0 deote the adom gaph pocess. Set M = τ( G; δ 4, let G = G M, Small = D l 0.9 (G, G = G[V \ Small] ad deote by the umbe of vetices i G. By Popositio. we ca assume that Small 0.. Settig c =, ad R = 9 40, the coditios of Lemma.5 ae met, ad thus thee exists a subgaph Ĝ G such that Ĝ M X 9 ad e(ĝ l , Agai, Make s stategy esembles that peseted i the pevious two cases, but ow it cosists of two phases. Let e i deote the edge selected by Make i his ith move ad let H i = (V, {e 1,..., e i } deote the gaph Make has built duig his fist i moves. Let H deote Make s gaph at the ed of the fist phase ad let H deote Make s gaph at the ed of the secod phase, i.e. Make s fial gaph. Befoe the game stats, Make splits the boad E(G ito thee pats F 1 = E(Ĝ, F = E G (Small, V \ Small ad F = E(G \ Ĝ. Duig the fist phase, Make plays two games i paallel, oe o F 1 ad the othe o F. Fo evey j 1, o his jth move of the fist phase, Make claims a edge of F 1 F, accodig to his stategy fo each of the two games. If o his jth move Beake claims a edge of F i, fo some i {1, }, the Make claims a edge of F i as well (uless he has aleady achieved his goal i the game o F i. If Beake claims a edge of F, the Make claims a edge of F 1 F. Playig o the edges of F 1, Make aims to build a (9 /40, -expade, H 1. As oted above, Make has a wiig stategy fo this game. Moeove, sice F 1 l 0.97, Make ca build such a expade withi at most t 1,1 := l 0.97 moves. Playig o the edges of F, Make follows a simple paiig stategy which guaatees that, by the ed of the game, the gaph, H, which Make costucts, will satisfy d H (v fo evey v Small. To achieve this goal, wheeve Beake claims a edge which is icidet with some vetex v Small, Make espods by claimig a diffeet edge icidet with v, uless his cuet gaph aleady cotais two edges which ae icidet with v

11 i which case he claims aothe fee edge of F 1 F which bigs him close to his goal i the coespodig game. Hece, the umbe of moves equied fo Make to each his goal i the game o F is at most t 1, := Small 0.. It follows by Popositio.4 that Small is a idepedet set ad that o two edges emittig fom Small ae icidet with the same vetex of V \ Small. Hece, Make s gaph, H, satisfies N H (U U fo evey U Small. Applyig Popositio. ad otig that 9 /40 /5, it follows that H = H 1 H is a (/5, -expade. Clealy, Make s fial gaph H is a (/5, -expade as well. A cucial poit to keep i mid is that the umbe of moves equied fo Make to costuct his (/5, - expade H, is t 1 = t 1,1 + t 1, = o( l Afte havig completed the costuctio of H, Make poceeds to the secod phase of his stategy. Let t deote the umbe of moves Make plays duig the secod phase. Fo evey t 1 < j t 1 + t, o his jth move, Make claims a edge of G which is a booste with espect to H j 1. This is possible sice, thoughout the game Beake claims at most t 1 +t t 1 + edges of G, but by Lemma.4, w.h.p. eithe H j 1 is Hamiltoia o it has at least l 0.98 > t 1 + boostes amog the edges of G. It follows by the defiitio of a booste that eithe H j is Hamiltoia o l(h j > l(h j 1. Repeatig the same agumet t times, we coclude that H is Hamiltoia as claimed. 4 Poof sketch of Theoems 1.4 ad 1.5 We ow sketch how the poof of Theoem 1. ca be adapted so as to etail Theoem 1.5. Similaly, the poof of Theoem 1.4 ca be obtaied usig appopiate modificatios to the poof of Theoem 1., but as this case is simple, we omit the details. It suffices to pove that whe emovig all vetices of degee at most l 0.9 fom the adom gaph G(, M, whee M = τ( G; δ 4k, playig o this subgaph G o vetices, w.h.p. Make ca quickly (that is, withi o( l moves build a (9 /40k, k- expade H fo which the popety M with = /l 0.4 holds. Moeove, at the same time, Make ca esue that the miimum degee of his gaph will be at least k. Afte the emoval of 0 i k 1 edge-disjoit Hamilto cycles fom the oigial gaph we have emoved a i-egula gaph fom fom H ad ae left with a gaph Ĥi (which is spaed by the vetices which ae ot i Small fo which (U NĤi k U i U (k + U fo evey U V (H of cadiality U 9 /40k. To complete the poof it is left to ote that fo the choice of the paamete guaatees that betwee sets of liea size thee is a supe-liea umbe of edges. It is ot had to see that addig back the vetices of Small who ae all icidet to at least k i edges esults i a coected (/5, -expade. This gaph has may boostes which Beake could ot have take them all, ad Make ca thus cotiue playig fo aothe Hamilto cycle usig the boostes left i the gaph. As thee is a supe-liea umbe of boostes ad Beake ca claim at most of them pe Hamilto cycle, Make ca keep playig this way util he completely satuates his vetices of miimum degee. 5 Cocludig emaks ad ope poblems I this pape we have poved the hittig time esults fo the pefect matchig, k-vetex-coectivity, ad Hamiltoicity Make-Beake games alog with some geealizatios. These esults futhe exemplify the so-called miimum degee pheomea which occus i adom gaphs, whee the bottleeck fo a typical adom gaph to satisfy some (global popety is the existece of vetices whose degee is too small fo the popety to hold. The mai igediet of ou poofs is to ote that oce the vetices of small degee ae take cae of the emaide of the adom gaph is ich eough (ad, actually, much iche is eeded so that we ca deduce the equied popeties. This settig, i a sese, emphasizes the tue bottleeck as the oe just descibed. I the cotext of adom egula gaphs thee ae o special vetices to take cae of, ad the poblem becomes much moe iticate. Let G,d deote the uifom pobability space of all adom egula gaphs o a fixed set of vetices. A iteestig questio to tackle would be Poblem. What is the miimal d fo which w.h.p. G,d M HAM? Classical esults fo this mode imply that w.h.p. G,d HAM fo all fixed d, but it follows fom a esult of Hefetz et. al [15] that d must at least 5 fo G,d M HAM. The fist ad fouth autho with Sudakov [4] poved that this miimal d is fixed, but to get the miimal value equies futhe ideas. Ackowledgmets This eseach was patially coducted while the authos wee peset (as guests o membes at the Istitute of Theoetical Compute Sciece at ETH Züich. We would like to thak Agelika Stege ad he goup fo the suppot ad wodeful facilities povided duig this time. Refeeces [1] N. Alo ad J. H. Spece. The Pobabilistic Method. Wiley-Itesciece Seies i Discete Mathematics ad Optimizatio. Joh Wiley & Sos, thid editio, 008.

12 [] J. Beck. O positioal games. Joual of Combiatoial Theoy, Seies A, 0(:117 1, [] J. Beck. Combiatoial Games: Tic-Tac-Toe theoy. Cambidge Uivesity Pess, New Yok, 008. [4] S. Be-Shimo, M. Kivelevich, ad B. Sudakov. Local esiliece ad Hamiltoicity Make-Beake games i adom egula gaphs. Combiatoics, Pobability, ad Computig, to appea. [5] B. Bollobás. The evolutio of spase gaphs. I B. Bollobás, edito, Poceedigs of Cambidge Combiatoial cofeece i hoo of Paul Edős, Gaph Theoy ad Combiatoics, pages Academic Pess, [6] B. Bollobás. Radom Gaphs. Cambidge Uivesity Pess, 001. [7] B. Bollobás ad A. Fieze. O matchigs ad hamiltoia cycles i adom gaphs. I Radom Gaphs (Pozań 198, volume 8 of Aals of Discete Mathematics, pages 46. Noth-Hollad, Amstedam, [8] B. Bollobás ad A. G. Thomaso. Radom gaphs of small ode. I Radom Gaphs (Pozań 198, volume 8 of Aals of Discete Mathematics, pages Noth-Hollad, Amstedam, [9] V. Chvátal ad P. Edős. Biased positioal games. I Algoithmic aspects of combiatoics (Vacouve 1976, volume of Aals of Discete Mathematics, pages [10] P. Edős ad J. Selfidge. O a combiatoial game. Joual of Combiatoial Theoy, Seies A, 14:98 01, 197. [11] O. N. Feldheim ad M. Kivelevich. Wiig fast i spase gaph costuctio games. Combiatoics, Pobability ad Computig, 17(6: , 008. [1] A. Fieze ad M. Kivelevich. O two Hamilto cycle poblems i adom gaphs. Isael Joual of Mathematics, 166:1 4, 008. [1] D. Hefetz, M. Kivelevich, M. Stojaković, ad T. Szabó. Fast wiig stategies i Make-Beake games. Joual of Combiatoial Theoy, Seies B, 99(1:9 47, 009. [14] D. Hefetz, M. Kivelevich, M. Stojaković, ad T. Szabó. A shap theshold fo the Hamilto cycle Make-Beake game. Radom Stuctues ad Algoithms, 4(1:11 1, 009. [15] D. Hefetz, M. Kivelevich, M. Stojaković, ad T. Szabó. Global Make-Beake games o spase gaphs. Euopea Joual of Combiatoics, to appea. [16] D. Hefetz ad S. Stich. O two poblems egadig the Hamilto cycle game. The Electoic Joual of Combiatoics, 16(1:R8, 009. [17] S. Jaso, T. Luczak, ad A. Ruciński. Radom Gaphs. Wiley-Itesciece Seies i Discete Mathematics ad Optimizatio. Joh Wiley & Sos, 000. [18] J. Komlós ad E. Szemeédi. Limit distibutios fo the existece of Hamilto cicuits i a adom gaph. Discete Mathematics, 4(1:55 6, 198. [19] A. Lehma. A solutio of the Shao switchig game. Joual of the Society fo Idustial ad Applied Mathematics, 1(4:687 75, [0] E. M. Palme ad J. J. Spece. Hittig time fo k edge-disjoit spaig tees i a adom gaph. Peiodica Mathematica Hugaica, 1(:5 40, [1] A. Pekeč. A wiig stategy fo the Ramsey gaph game. Combiatoics, Pobability ad Computig, 5(:67 76, [] M. Stojaković. Games o Gaphs. PhD thesis, ETH Züich, 005. [] M. Stojaković ad T. Szabó. Positioal games o adom gaphs. Radom Stuctues ad Algoithms, 6(1-:04, 005. [4] D. B. West. Itoductio to Gaph Theoy. Petice Hall, 001. A Pelimiaies I this sectio we cite some tools which we will make use of i the succeedig sectios. Fist, we will eed to employ bouds o lage deviatios of adom vaiables. We will mostly use the followig well-kow boud o the lowe ad the uppe tails of the Biomial distibutio due to Cheoff (see e.g. [1, Appedix A]. Theoem A.1. (Cheoff bouds If X B(, p the 1. P [X < (1 εp] < exp( ε p fo evey ε > 0;. P [X > (1 + εp] < exp( p fo evey ε 1. It will sometimes be moe coveiet to use the followig boud o the uppe tail of the Biomial distibutio. Lemma A.1. If X Bi(, p ad k p, the P [X k] (ep/k k. Note that the boud give i Lemma A.1 is especially useful whe k is much lage tha p. A.1 Basic positioal games esults The followig theoem is a classical esult of Edős ad Selfidge [10] which povides a useful sufficiet coditio fo Beake s wi i the (X, F game. Theoem A.. (Edős ad Selfidge [10] Fo ay hypegaph (X, F, if A < 1, A F the Beake, playig as the fist o secod playe, has a wiig stategy fo the (X, F game. The followig simple lemma is useful whe a playe is tyig to esue expasio of small sets. A simila lemma appeaed i [14].

13 Lemma A.. Fo evey itege k > 0, if H is a gaph o vetices with miimum degee δ(h 5k, the H M δk. Moeove, Make ca wi the miimum degee k game o the edge set of H i at most k moves. Poof. We defie a ew gaph H, whee H = H if all the degees i H ae eve, ad othewise H is the gaph obtaied fom H by addig a ew vetex v ad coectig it to evey vetex of odd degee i H. Sice all degees of H ae eve, it admits a Euleia oietatio H. Fo evey v V (H, let E(v = {{v, u} E(H : (v, u E( H }. Clealy, E(v d H (v/ d H (v/ 5k/ ad the sets {E(v} v V (H ae paiwise disjoit. I evey oud, if Beake claims a edge of E(v, the Make espods by claimig a edge of E(v \ {{v, v }}, uless he aleady has k edges icidet with v i which case Make poceeds by claimig a edge of E(u, whee u is some vetex such that Make did ot yet claim k of its icidet edges (if o such vetex exists, the the game was aleady wo by Make. Note that sice E(v / k, Make ca always play accodig to this stategy, ad is eve foced to pick a edge icidet with v. Hece, Make claims oly edges of the oigial gaph H. Disegadig the oietatio, afte at most k moves, the gaph spaed by Make s edges has miimum degee at least k as claimed. B Popeties of adom gaphs ad adom gaph pocesses We stat with a vey simple claim egadig the umbe of edges i the Biomial adom gaph model G(, p whose poof is stadad ad is theefoe omitted. l Popositio B.1. If p, the w.h.p. e(g(, p p. The followig estimates the pobability of a vetex to be i D t (G(, p. Popositio B.. Fo evey itege t l 0.9 ad evey vetex v, if l < p < l, P [v D t (G(, p] 1+o(1. < l 0.9 o(1 1+o(1 < 1+o(1. Fo evey fixed itege k 1 we defie two fuctios as follows: ( l + (k 1 l l l l l m k = ; ( l + (k 1 l l + l l l M k =. The followig lemma (see e.g. [6] gives a pecise behavio of the miimum degee of the adom gaph pocess. Lemma B.1. Fo evey fixed itege k 1, if G is the adom gaph pocess the w.h.p. m k < τ( G; δ k < M k. Lastly, we state some stuctual popeties of the the adom gaph model G(, M, which we have see to be equivalet to stoppig the adom gaph pocess at G M. Popositio B.. Fo evey fixed itege k, if G = {Gi } ( i=0 is the adom gaph pocess ad M = τ( G; δ k the w.h.p. G M satisfies that e G (U < U l 0.8 fo evey subset of vetices U V of cadiality 1 U l 0.. Popositio B.4. Fo evey fixed iteges k 1, ad = l 0.4 if G = {Gi } ( i=0 is the adom gaph pocess ad M = τ( G; δ k the w.h.p. e GM (U, W l 0.1 fo evey disjoit pai of subsets of vetices U, W V of cadiality U = W =. I ode to pove these esults oe ca esot to the use of G(, p, whee the aalysis is much simple, ad use Popositio 1. to tasfe the esults to the oigial adom gaph model. The full poofs of these two popositios will appeas i the full vesio of this pape. Poof. Let G = (V, E G(, p, the fo evey vetex v V we have that d G (v Bi( 1, p, ad theefoe we have that fo evey itege t l 0.9 P [v D t (G] P [Bi( 1, p < t] ( 1 t p t (1 p 1 t t ( ep t t e p( 1 t t