The Game Chromatic Number of Random Graphs

Transcription

1 The Game Chromatic Number of Radom Graphs Tom Bohma, 1, * Ala Frieze, 1, Bey Sudakov 2,3, 1 Departmet of Mathematics, Caregie Mello Uiversity, Pittsburgh, Pesylvaia 15213; tbohma@math.cmu.edu; ala@radom.math.cmu.edu 2 Departmet of Mathematics, Priceto Uiversity, Priceto, New Jersey 08544; bsudakov@math.priceto.edu 3 Istitute for Advaced Study, Priceto, New Jersey Received 23 December 2005; accepted 8 August 2006 Published olie 17 April 2007 i Wiley IterSciece ( DOI /rsa ABSTRACT: Give a graph G ad a iteger k, two players take turs colorig the vertices of G oe by oe usig k colors so that eighborig vertices get differet colors. The first player wis iff at the ed of the game all the vertices of G are colored. The game chromatic umber χ g (G) is the miimum k for which the first player has a wiig strategy. I this study, we aalyze the asymptotic behavior of this parameter for a radom graph G,p. We show that with high probability, the game chromatic umber of G,p is at least twice its chromatic umber but, up to a multiplicative costat, has the same order of magitude. We also study the game chromatic umber of radom bipartite graphs Wiley Periodicals, Ic. Radom Struct. Alg., 32, , INTRODUCTION Let G = (V, E) be a graph ad let k be a positive iteger. Cosider the followig game i which two players Maker ad Breaker take turs colorig the vertices of G with k colors. Each move cosists of choosig a ucolored vertex of the graph ad assigig to it a Correspodece to: A. Frieze *Research supported i part by NSF grat DMS Research supported i part by NSF grat CCF Research supported i part by NSF CAREER award DMS , NSF grat DMS , USA-Israeli BSF grat, Alfred P. Sloa fellowship, ad the State of New Jersey Wiley Periodicals, Ic. 223

2 224 BOHMAN, FRIEZE, AND SUDAKOV color from {1,..., k} so that resultig colorig is proper, i.e., adjacet vertices get differet colors. Maker wis if all the vertices of G are evetually colored. Breaker wis if at some poit i the game the curret partial colorig caot be exteded to a complete colorig of G, i.e., there is a ucolored vertex such that each of the k colors appears at least oce i its eighborhood. We assume that Maker goes first (our results will ot be sesitive to this choice). The game chromatic umber χ g (G) is the least iteger k for which Maker has a wiig strategy. This parameter is well defied, sice it is easy to see that Maker always wis if the umber of colors is larger tha the maximum degree of G. Clearly, χ g (G) is at least as large as the ordiary chromatic umber χ(g), but it ca be cosiderably more. For example, let G be a complete bipartite graph K, mius a perfect matchig M ad cosider the followig strategy for Breaker. If Maker colors vertex v with color c the Breaker respods by colorig the vertex u matched with v i the matchig M with the same color c. Note that ow c caot be used o ay other vertex i the graph. Therefore, if the umber of colors is less tha, Breaker wis the game. This shows that there are bipartite graphs with arbitrarily large game chromatic umber ad thus there is o upper boud o χ g (G) as a fuctio of χ(g). The game was first cosidered by Brams about 25 years ago i the cotext of colorig plaar graphs ad was described i Marti Garder s colum [10] i Scietific America i The game remaied uoticed by the graph-theoretic commuity util Bodlaeder [3] re-iveted it. It has bee studied for various classes of graphs i recet years. Faigle, Ker, Kierstead ad Trotter [9] proved that the game chromatic umber of a forest is at most 4, ad that there are forests which require that may colors. The game chromatic umber of plaar graphs was studied by Kierstead ad Trotter [13], who showed that for such graphs the game chromatic umber is at most 33. Moreover they proved that ay graph embeddable o a orietable surface of geus q has game chromatic umber bouded by a fuctio of q. Several additioal results o χ g ad some related parameters were obtaied i [4, 6 8, 12, 14, 16, 17]. For a recet survey see Barticki, Grytczuk, Kierstead ad Zhu [2]. I this article, we study the game chromatic umber of the radom graph G,p. As usual, G,p stads for the probability space of all labeled graphs o vertices, where every edge appears idepedetly with probability p = p(). We assume throughout the article that the edge probability p 1 η, where η>0is a arbitrarily small, but fixed, costat. Defie b = 1 ad ote that log 1 p b x = log x log x = (1 + o(1)) for all x 1 ad p = o(1). Our first log b p result determies the order of magitude of the game chromatic umber of G,p. Theorem 1.1. (a) There exists K > 0 such that for ε>0 ad p (log ) Kε 3 / we have that whp 1 χ g (G,p ) (1 ε) log b p. (b) If α>2is a costat, K = max { 2α, α α 1 α 2} ad p (log ) K / the whp χ g (G,p ) α log b p. 1 A sequece of evets E occurs with high probability (whp) iflim P(E ) = 1 Radom Structures ad Algorithms DOI /rsa

3 THE GAME CHROMATIC NUMBER OF RANDOM GRAPHS 225 It is atural to compare our bouds with the asymptotic behavior of the ordiary chromatic umber of radom graph. It is kow by the results of Bollobás [5] ad Łuczak [15] that whp χ(g,p ) = (1 + o(1)). Thus our result shows that the game chromatic umber 2log b p of G,p is at least twice its chromatic umber, but up to a multiplicative costat has the same order of magitude. As already metioed, there are graphs whose game chromatic umber is much larger the the ordiary oe. Our ext theorem provides the existece of a large collectio of such graphs. Let B,p deote the radom bipartite graph with two parts of vertices where each of the 2 possible edges appears radomly ad idepedetly with probability p. We obtai the followig bouds o the game chromatic umber of this graph. Theorem 1.2. (a) If p 2/ the χ g (B,p ) 10(log )(log b p). (b) If α>2is a costat, K = max { 2α, α α 1 α 2} ad p (log ) K / the whp χ g (B,p ) α log b p. The rest of this article is orgaized as follows. The ext two sectios cotai proofs of lower ad upper bouds i Theorem 1.1. I Sectio 4, we cosider the game chromatic umber of radom bipartite graphs ad prove Theorem 1.2. The last sectio of the article cotais some cocludig remarks ad ope problems. Uless the base is specifically metioed, log will refer to atural logarithms. We ofte refer to the followig Cheroff-type bouds for the tails of biomial distributios (see, e.g., [1] or [11]). Let X = i=1 X i be a sum of idepedet idicator radom variables such that P(X i = 1) = p i ad let p = (p 1 + +p )/. The P(X (1 ε)p) e ε2 p/2, P(X (1 + ε)p) e ε2p/3, ε 1, P(X µp) (e/µ) µp. 2. LOWER BOUND ON THE GAME CHROMATIC NUMBER OF G,p Suppose that p (log ) Kε 3 /, where K is a sufficietly large costat, ad that the umber of colors k satisfies k (1 ε). We begi by defiig a series of umbers (which will log b p serve as cut-offs for Breaker s strategy). Let l 1 = log b log b log b p 10 log b log ad ote that l 1 has bee chose so that we have (1 p) l 1 = log b p(log ) 10 = (1 + o(1)) l 1(log ) 10. (1) Radom Structures ad Algorithms DOI /rsa

4 226 BOHMAN, FRIEZE, AND SUDAKOV Set l 2 = εl 1 20 ad l 3 = ε3 l For S [] let N(S) be the eighbors of S which are ot i S ad let N(S) =[]\(S N(S)). We are ow ready to describe Breaker s strategy. Fix a color i. Wheever Maker uses i, Breaker will respod by usig color i i his ext move. At the begiig Breaker chooses this vertex arbitrarily; oly whe l 1 vertices are colored i will Breaker choose carefully the ext vertex to color. Let T deote the set of ucolored vertices i N(C i ) at the time whe the set of vertices C i that have bee colored with i satisfies C i =l 1. At this poit Breaker idetifies a maximum size idepedet subset I 1 of T. Whe Breaker ext uses color i, he will color the vertex v T, which has as may eighbors i I 1 as possible. After this, I 1 I 1 \ N(v). Whe I 1 l 3 we say that Breaker has completed elimiatio iteratio 1. After completig elimiatio iteratio j, Breaker will start a ew iteratio by idetifyig the largest idepedet set I j+1 i the set of ucolored vertices i the curret N(C i ) ad cotiue with the previous strategy. This cotiues as log as at the start of a ew iteratio the set N(C i ) cotais a idepedet set of ucolored vertices of size at least l 2. Oce N(C i ) does ot cotai ay more idepedet sets of size l 2, from the o Breaker agai colors arbitrarily with i whe desired. To validate Breaker s strategy we must establish a few facts. We begi by cosiderig the size of N(C i ) whe C i =l 1. Lemma 2.1. For every subset S [] of size S =l 1 whp l 1 (log ) 9 N(S) l 1 (log ) 11. Proof. Fix S with S =l 1. The size of N(S) is distributed as the biomial B( l 1, (1 p) l 1). Therefore by (1) the expected size of N(S) is (1 + o(1))l 1 (log ) 10. Thus, it follows from the Cheroff bouds that P [ S : S =l 1, N(S) / [l 1 (log ) 9, l 1 (log ) 11 ] ] ( ( 2 )e l 1 (log ) 10) = o(1). Next we ote that if the umber of ucolored vertices i N(C i ) is sufficietly large, the Breaker should be able to choose a vertex that reduces the size of I j by a substatial amout. l 1 Lemma 2.2. Whp there do ot exist S, A, B [] such that 1. S =l 1,a= A [l 3,3l 1 ], B b 1 = 100ε 1 l 1 (log ) A, B N(S) ad A B =. 3. Every x B has fewer tha ap/2 eighbors i A. Proof. Applyig Lemma 2.1 to boud the size of N(S) ad usig l 3 p = (K log log ) we see that the probability of this evet is at most ( o(1) + l 1 ) 3l 1 a=l 3 o(1) + l 1 = o(1). ( )( ) l1 (log ) 11 l1 (log ) 11 P(B(a, p) ap/2) b 1 a 3l 1 b 1 a=l 3 (log ) 11(a+b1) e ab1p/8 Radom Structures ad Algorithms DOI /rsa

5 THE GAME CHROMATIC NUMBER OF RANDOM GRAPHS 227 Now we cosider the umber of elimiatio iteratios. Defie l 2 = εl 1 21, ad ote that we have l 2 <l 2 l 3. Each elimiatio iteratio removes at least l 2 vertices from the set of vertices that ca be colored with color i. Note further that these sets are disjoit ad that each forms a idepedet set i our graph. Lemma 2.3. Whp there do ot exist S, T 1, T 2,..., T a1 [],a 1 = 2000ε 2 such that 1. S, T 1, T 2,..., T a1 are pair-wise disjoit idepedet sets. 2. S =l T i =l 2, i = 1, 2,..., a N(S) T i =, i = 1, 2,..., a 1. Proof. Let E 1 be the evet that such a collectio of sets exists. The, usig l 1 /l 2 = (1), l 1 = O( ) log p (log ) Kε 3 /2 together with (1) ad Lemma 2.1, we have ( )( ) l1 (log ) 11 a1 P(E 1 ) o(1) + (1 p) a 1( l 2 ) 2 l 1 l 2 ( ) e l1 ( ) e l1 (log ) 11 a1 l 2 o(1) + (1 p) (l 2 1)/2 l 1 l 2 ( ) e l1 ( ) o(1) + ((log ) 12 l1 (log ) 10 ε/42 ) a1 εl 1 /21 l 1 ( ) l1 ( ) a1 ε 2 l 1 /1000 l1 o(1) + (log ) a 1 εl 1 l 1 ( l1 o(1) + (log )2000/ε = o(1). We will complete the proof by showig that most colors are used o roughly l 1 vertices. We will use the followig Lemma to boud the umber of colors that are used o sigificatly more vertices. We defie a fourth cut-off ) l1 l 0 = l l 1 l 3 p a 1. Note that a 1 l 1 /l 3 is a costat ad (1 p) l 0 = ((1 p) l 1). Lemma 2.4. Whp there do ot exist pair-wise disjoit sets S 1, S 2,..., S b2, U, b 2 = l 1 (log ) 7 such that 1. S i =l 0 for i = 1, 2,..., b U = / log. 3. U N(S i ) l 1 (log ) 8 for i = 1, 2,..., b 2. Radom Structures ad Algorithms DOI /rsa

6 228 BOHMAN, FRIEZE, AND SUDAKOV Proof. Let E 2 be the evet that such a collectio of sets exists. For every choice of S i ad U the size U N(S i ) is distributed as the biomial B(/ log, (1 p) l 0) with expectatio (/ log )(1 p) l 0 = ((1 p) l 1/ log ) = (l 1 (log ) 9 ). Thus, by the Cheroff boud, ( )(( P(E 2 ) )P ( B(/ log, (1 p) l 0) l 1 log 8 )) b 2 / log l 0 / log 2b 2 l 1e (b 2 l 1 (log )9 ) = o(1). We ow use these Lemmas to complete the proof, assumig that the associated low probability evets do ot occur. Assume for the sake of cotradictio that the game reaches the poit where oly / log vertices remai to be colored. Let U be the set of ucolored vertices. Let C i deote the set of vertices colored i at this poit ad let c i = C i for i = 1, 2,..., k. Observe that whp c i (2 +.01ε)l 1, i = 1, 2,..., k (2) sice the right had side is a upper boud o the size of a idepedet set i G,p. Claim 2.5. Let i be a color such that c i (1 + ε/4)l 1.IfC i are the first l 0 vertices to be colored with color i the we have N ( ) C i U b1 = 100l 1(log ) 2 ε Proof. Assume for the sake of cotradictio that N(C i ) U > b 1. Let S t be the set of vertices which are colored i at time t. At all times t such that S t <l 0 the set of ucolored vertices i N(S t ) has size at least N(C i ) U > b 1. Let I j be the idepedet set that is beig elimiated at time t. Sice I j is smaller tha the idepedece umber of G,p, Lemma 2.2 implies that Breaker ca choose a vertex that elimiates at least I j p/2 vertices from I j. Therefore, each elimiatio iteratio ivolves at most 2 I j /(l 3 p/2) <9l 1 /(l 3 p) uses of color i. But Lemma 2.3 implies that there are at most a 1 elimiatio iteratios. Therefore, Breaker will complete all of the elimiatio iteratios before color i has bee used l 0 times. Note that after the elimiatio process is completed oe ca oly color at most l 2 = εl 1 /20 vertices by color i. Therefore, This is a cotradictio. c i <l 1 + 9a 1 l 1 /(l 3 p) + εl 1 /20 <(1 + ε/4)l 1. It follows from Claim 2.5 ad Lemma 2.4 that there at most b 2 = colors i such l 1 (log ) 7 that c i >(1 + ε/2)l 1. Applyig this fact together with (2) ad k (1 ε)/l 1 we obtai log = k c i b 2 (2 +.01ε)l 1 + (k b 2 )(1 + ε/4)l 1 <(1 ε/2). i=1 This is a cotradictio. Radom Structures ad Algorithms DOI /rsa

7 THE GAME CHROMATIC NUMBER OF RANDOM GRAPHS UPPER BOUND ON THE GAME CHROMATIC NUMBER OF G,p Let α be ay costat greater tha 2, K > max{ 2α, α }, p >(log α 1 α 2 )K / ad let the umber of colors be k = α. We begi with Maker s strategy. Let C = (C log b p 1, C 2,..., C k ) be a collectio of pair-wise disjoit sets. Let C deote k i=1 C i. For a vertex v let ad set A(v, C) = {i [k] : v is ot adjacet to ay vertex of C i }. a(v, C) = A(v, C). Note that A(v, C) is the set of colors that are available at vertex v whe the partial colorig is give by the sets i C ad v C. Maker s strategy ca ow be easily defied. Give the curret color classes C, Maker chooses a ucolored vertex v with the smallest value of a(v, C) ad colors it by ay available color. To establish that Maker s strategy succeeds whp, we cosider a sequece of ladmarks i the play of the game. As the game evolves, we let u deote the umber of ucolored vertices i the graph. So, we thik of u as ruig backward from to 0. Below we defie a sequece of thresholds d 0 d 1 d r+1 ad cosider the times u i, which are defied to be the last times (i.e. miimum value of u) for which Maker colors a vertex for which there are at least d i available colors. We begi with the first ladmark, u 0. Let β = k (p) 1/α = α (p) 1/α log b p, 10 log γ = β ad B(C) = {v : a(v, C) <β/2}. We begi by showig that with high probability every colorig of the full vertex set has the property that there are at most γ vertices with less tha β/2 available colors. Lemma 3.1. Whp, for all collectios C, B(C) γ. Proof. Fix C. The for every v / C, the umber of colors available at v is the sum of idepedet idicator variables X i, where X i = 1ifv has o eighbors i C i. The P(X i = 1) = (1 p) Ci ad sice (1 p) t is a covex fuctio we have E(a(v, C)) = It follows from the Cheroff boud that k (1 p) C i i=1 k(1 p) ( C C k )/k k(1 p) /k = β. P(a(v, C) β/2) e β/8. Radom Structures ad Algorithms DOI /rsa

8 230 BOHMAN, FRIEZE, AND SUDAKOV Thus, P( C with B(C) >γ) k ( γ ) e βγ/8 = o(1). Set u 0 to be the last time for which Maker colors a vertex with at least d 0 = β/2 available colors, i.e., u 0 = mi {u : a(v, C u ) d 0 = β/2, for all v } C u, where C u deotes the collectio of color classes whe u vertices remai ucolored. It follows from Lemma 3.1 that whp u 0 γ (we apply the Lemma to the fial colorig). This implies that at some poit where the umber of ucolored vertices is less tha γ, every vertex still has at least d 0 = β/2 available colors. I particular, if β/2 >γ(this happes, e.g., for costat p ad α>2) we see that Maker wis the game sice o vertex will ever ru out of colors. O the other had, the proof that Maker s strategy succeeds also for p = o(1) eeds more delicate argumets, which we preset ext. Sice u 0 is both defied ad uderstood, we are ready to defie u 1,..., u r+1. These ladmarks are defied i terms of thresholds d 1,..., d r+1, where d i is a lower boud o the umber of colors available at every ucolored vertex (ote that d 0 = β/2 was set above). We set d i+1 = d i x i where the x i s will be defied below ad u i = mi {u : a(v, C u ) d i, for all v } C u We will choose the x i s so that u i log log implies u i+1 u i 10. (3) We defie r (ad hece establish the ed of our series of ladmarks) by u r log log > u r+1. Note that if (3) holds the we have We will also esure that we have r log. r x i d 0 /2. (4) i=0 If we ca choose the x i s so that (3) ad (4) are satisfied the Maker will succeed i wiig the game. Ideed, whe there are u r ucolored vertices, there are fewer tha 10 log log vertices to color ad the lists of available colors at these vertices have size at least r d 0 x i d 0 /2 >β/4 > 10 log log, (5) i=0 where the lower boud follows from the facts that β ( (p) α 1 ) α / log p, p >(log ) K ad K > 2α. α 1 The key to our aalysis (ad the choice of x i s) is the followig observatio. Betwee the poit whe there are u i ucolored vertices ad the ed of the game every vertex i []\ C ui+1 must lose at least d i d i+1 = x i of its available colors. Ideed, such a vertex v must have at least d i available colors whe there are u i vertices ucolored but has less the d i+1 available colors whe v itself is colored. This implies that the graph iduced o Radom Structures ad Algorithms DOI /rsa

9 THE GAME CHROMATIC NUMBER OF RANDOM GRAPHS 231 []\ C i has at least u i+1 x i edges. Before we proceed, we eed aother techical Lemma, which bouds the umber of edges spaed by subsets of G,p. For each positive iteger s defie φ = φ(s) = (5ps + log )s. Lemma 3.2. edges. Proof. Whp every subset S of G,p of size s spas at most φ = φ(s) = (5ps+log )s P( S with e(s) >φ) ( )(( s 2) s φ s=2 ( e s s=2 = o(1). ) p φ ( e ) ) log s 10 We, heceforth, assume that the low probability evets give i Lemmas 3.1 ad 3.2 do ot occur. It follows from our key observatio that we have Thus to achieve (3) it suffices to take x i u i+1 φ(u i ) = (5pu i + log )u i. (6) x i 10(5pu i + log ). But, sice u i u 0 /10 i ad u 0 γ, we ca take x i = 5pγ + 10 log. 10i 1 Checkig (4) we see that we require r x i 60pγ + 10(log ) 2 i=0 to be less tha d 0 /2, ad so we eed to verify 600p log β + 10(log ) 2 β 4. (7) Note that (5) follows immediately from (7). Sice p >(log ) K, K > max { 2α, α α 1 α 2} ad α>2we have that ) ((p) α 1 α β max { (log ) 2, (p) 1/α log }, log p which implies (7). Radom Structures ad Algorithms DOI /rsa

10 232 BOHMAN, FRIEZE, AND SUDAKOV 4. PROOF OF THEOREM 1.2 We start by provig part (a) of Theorem 1.2. Suppose that 2/ p 1 η ad the umber of colors is at most k =. Also recall that if p = o(1) the log 10(log )(log b p) b x = (1+o(1)) log x. p Breaker employs the followig strategy. He chooses oe part of the bipartite graph, which we deote by W B to be Breaker s side ad thiks of the opposite part W M as Maker s side. Loosely speakig, Breaker tries to elimiate the colorig possibilities o Maker s side. To state Breaker s strategy precisely, we itroduce a defiitio. We say that a color is dead if it is available o less tha 6(log )(log b p) vertices o Maker s side W M. Breaker colors accordig to the followig three simple rules: 1. Oly color o Breaker s side W B, 2. Do ot use a dead color, ad 3. If possible, respod to a move by Maker o Maker s side i kid (i.e., whe Maker plays o Maker s side with a particular color the Breaker s first choice is to play the same color o Breaker s side). We say that a color escapes if it is ot dead ad Breaker stops playig this color because he caot choose a vertex o his side that ca be colored with this color. Note that it follows from the third rule for Breaker that the umber of times a color is played o Breaker s side is at least the umber times it is played o Maker s side as log as the color is either dead or has escaped. Note further that there may be rouds whe Breaker s move will ot be dictated by the rules above. Durig these rouds Breaker simply colors arbitrarily o Breaker s side. We cotiue play util every color either dies or escapes; that is, we play util Breaker caot follow his colorig rules. Suppose that this happes after ν M ν B vertices have bee colored o Maker s ad Breaker s sides respectively. We will show that whp Breaker will be i a wiig positio by this time. Recall that W M, W B deote Maker ad Breakers sides of the bipartitio. For X W B we let N(X) deote the set of vertices i W M that have o eighbors i X (i.e. N(X) = W M \ N(X)). Let λ 0 = log b log b log log b log b p log b 3 ad ote that λ 0 < log b p ad (1 p) λ 0 = 3(log )(log b p). (8) Lemma 4.1. Whp every subset L W B of size l λ 0 has at most 2(1 p) l oeighbors i W M. Proof. Fix L W B with L =l. The umber of o-eighbors of L i W M is distributed as the biomial B(, (1 p) l ). Thus, by the Cheroff bouds, λ 0 ( ) λ 0 ( e ) l P( L : N(L) 2(1 p) l ) e (1 p)l /3 e (1 p) l /3. l l Now if l λ 0 < log b p the l=1 log( l e (1 p)l /3 ) = l log (1 p) l /3 l=1 { if l log 0 if. log l λ 0 Radom Structures ad Algorithms DOI /rsa

11 THE GAME CHROMATIC NUMBER OF RANDOM GRAPHS 233 Therefore, P( L : N(L) 2(1 p) l ) log l=1 ( e l ) l e + λ 0 l= log ( e l ) l = o(1). It follows from (8), Lemma 4.1 ad the defiitio of a dead color that Breaker makes a rule based use of each color at most λ 0 times. Usig the fact that at least as may vertices will be colored o Breaker s side as o Maker s side ad that they both had the same umber of turs we coclude that the umber of colored vertices at the poit whe Breaker stops satisfies ν M ν B 2kλ 0 5 log. Let c 1,..., c t be the colors that escape. Let M i, B i be the sets of vertices with color c i o Maker s ad Breaker s sides respectively at the momet that Breaker stops playig color c i because he is forced to by the rules. Let m i = M i ad set α = t (1 p) m i. i=1 Note first that m i b i = B i λ 0, i = 1, 2,..., t. Furthermore, because b i λ 0 we see that N(B i ) 2(1 p) b i 2(1 p) m i. We cosider two cases. Case 1. α<1/6. The total umber of vertices that ca be colored o Maker s side is at most the sum of (i) the umber of vertices colored so far, (ii) the umber of vertices that ca be colored with dead colors, ad (iii) the umber of vertices that ca be colored with escaped colors. Hece the umber of vertices that ca be colored o Maker s side is at most ν M + k 6(log )(log b p) + 2 o() + t (1 p) m i i=1 10(log )(log b p) 6(log )(log b p) + 3 <, ad therefore Maker ca ot complete the colorig of the graph. Case 2. α 1/6. I this case whp we arrive at a cotradictio. Let Z be the set of ν B vertices i W B that have bee colored so far. We have Z =ν B /(5 log ). Whe a color c i escapes it is uavailable to vertices o Breaker s side. It follows that all vertices i Y = W B \ Z have at least oe eighbor i M i. Radom Structures ad Algorithms DOI /rsa

12 234 BOHMAN, FRIEZE, AND SUDAKOV Let E be the evet that we have such a cofiguratio i.e. t small sets, whose eighborhoods each covers almost all of W B. Fix the sets M 1,..., M t ad Y. Sice Y =(1 o(1)), the probability that this collectio of sets satisfies the coditio is ( t Y { } t (1 (1 p) m i )) exp Y (1 p) m i e /7. i=1 The probability of the existece of ay such cofiguratio i our radom model is at most k t=1 ( λ0 ( ) )t ( ) e /7 λ 0 k e o() e /7 e /10+o() /7 = o(1) l /(5 log ) l=0 i= Upper Boud i Theorem 1.2 The proof here is essetially the same as for Theorem 1.1(a). Let the vertex bipartitio be deoted V 1, V 2. Aside from modifyig the statemet of Lemma 3.1 to say that the sets i C are cotaied i V i ad v V 3 i, the proof goes through basically uchaged. 5. CONCLUDING REMARKS AND OPEN PROBLEMS I this article, we obtai upper ad lower bouds o the game chromatic umber of G,p which differ oly by a multiplicative costat. It would be very iterestig to improve our result ad determie the asymptotic value of this parameter for radom graphs. Our results suggest that i fact the followig should be true. Cojecture 5.1. If p 1 η for some costat η>0ad p, the whp χ g (G,p ) = (1 + o(1)) log b p, where b = 1/(1 p). We cojecture that the game chromatic umber of radom bipartite graph B,p has the same order of magitude whp. We did ot succeed i provig the correct lower boud. As oe fial remark, there has bee much work doe o the cocetratio of the the chromatic umber of G,p. Noe of this is applicable to χ g (G,p ).Nowχ g (G,p ) should also be cocetrated, but provig this may require some ew approaches to provig cocetratio. REFERENCES [1] N. Alo ad J. H. Specer, The probabilistic method, 2d ed., Wiley, New York, [2] T. Barticki, J. A. Grytczuk, H. A. Kierstead, ad X. Zhu, The map colorig game, America Mathematical Mothly (i press). [3] H. L. Bodlaeder, O the complexity of some colorig games, Iterat J Foud Comput Sci 2 (1991), Radom Structures ad Algorithms DOI /rsa

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