THE DIAMETER GAME JÓZSEF BALOGH, RYAN MARTIN, AND ANDRÁS PLUHÁR

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1 THE DIAMETER GAME JÓZSEF BALOGH, RYAN MARTIN, AND ANDRÁS PLUHÁR Abstract. A large class of the so-called Positioal Games are defied o the complete graph o vertices. The players, Maker ad Breaker, take the edges of the graph i turs, ad Maker wis iff his subgraph has a give usually mootoe property. Here we itroduce the d-diameter game, which meas that Maker wis iff the diameter of his subgraph is at most d. We ivestigate the biased versio of the game; i.e., whe the players may take more tha oe, ad ot ecessarily the same umber of edges, i a tur. The 2-diameter game has the property that Breaker wis the game i which each player chooses oe edge per tur, but Maker wis as log as he is permitted to choose 2 edges i each tur whereas Breaker ca choose as may as /8 /8 /l /2. I additio, we ivestigate d-diameter games for d >. The diameter games are strogly related to the degree games. Thus, we also provide a geeralizatio of the fair degree game for the biased case.. Itroductio A so-called positioal game ca be viewed, i the most geeral settig, as a game i which two players Maker ad Breaker occupy vertices i a hypergraph. Maker wis if he ca occupy all vertices of oe hyperedge. Breaker wis if he ca occupy at least oe vertex of each hyperedge. For more iformatio o these games, see the excellet survey icluded i [4]. At each tur, let Maker make a moves ad Breaker make b. I this paper we let Maker start the game. We call such games a : b-games. If a = b, the game is fair, otherwise it is biased. If a = b >, the game is accelerated. I [4], Beck asked about the behavior of accelerated versus uaccelerated games. The specific example is oe i which players alterately take cois of the same type ad place them o a circular table so that the cois do ot overlap. The player who, for the first time, caot make a proper move, loses. I the : -game, Maker ca wi by placig a coi i the ceter ad make moves that reflect the secod player s move. Beck observed, however, that it is ot clear that Maker has a wiig strategy for the 99 Mathematics Subject Classificatio. 05C65, 9A43, 9A46. Key words ad phrases. Positioal games, radom graphs, diameter. The first author s research is partially supported by NSF grat DMS ad by OTKA grats T ad T The secod author s research is partially supported by NSA grat H The third author s research is partially supported by OTKA grats T ad T

2 2 BALOGH, MARTIN, AND PLUHÁR 2 : 2-game. Beck also observed that the arithmetic progressio game APm has the same property. Both questios are ope. It is well-studied i the literature that a game may have completely differet outcomes if it is played as : or a : b game, where either a or b is greater tha. Two classical results are due to Beck [] ad Chvátal ad Erdős [6] i our paper see Theorems 5 ad 6. Other variats of biased games were ivestigated by Pluhár [8, 9] ad Siebe [0]. Here, we ivestigate the so-called 2-diameter game ad i particular we prove that i the : -game, Breaker wis; but, i the 2 : 2-game, Maker wis. Let K the complete graph o vertices be the board. Maker wis if some give usually mootoe graph property P holds for the subgraph of his edges. A importat guide to uderstadig such games is the so-called probabilistic ituitio, for more details ad examples see [3]. I the probabilistic ituitio, we substitute the perfect players with radom players. It meas that we cosider the radom graph G, p with p = a/a+b; i.e., the edge distributio i the radom graph is the same as i the game. It was demostrated i a umber of cases that Maker wis if P holds, while Breaker wis if P does ot hold for G, p almost surely, see e.g., the papers [3, 5, ]. Here the property P d is that the graph has diameter at most d. We deote the correspodig d-diameter game by D d a : b, or more briefly, by D d if a = b =. Note that while it has bee kow that acceleratio may chage the outcome, here it is show that the : ad the 2:2-game have completely differet outcomes. Moreover this is the first o-trivial case i which the probabilistic ituitio fails completely i the :-game, ad is at least partially restored i the 2:2-game. Note further that this is the first o-artificial case that it is show that the : ad the 2 : 2-game have a differet outcomes, whe the miimum size of a wiig set of both players is large say, at least. The first o-trivial case is whe d = 2. Sice the diameter of G, /2 is 2 almost surely, oe would expect that Maker wis easily the game D 2. Nevertheless, the probabilistic ituitio fails! Propositio. If 3, the Maker wis the game D 2. If 4, the Breaker has a wiig strategy for the the game D 2. A little acceleratio of the game chages the outcome completely. I the case where a = 2, we expect the probabilistic ituitio to work; we prove a weaker result. Theorem 2. Maker wis the game D 2 2 : 8 /8 /log /2, ad Breaker wis the game D 2 2 : 2 + ǫ / l for ay ǫ > 0, provided is large eough. Before dealig with the geeral case, it is worth discussig the problem D 3 : b. Note that the threshold of the property P 3 is about /3 ; i.e., G, p has property P 3 with probability close to if p = /3+ǫ, ad it does ot have property P 3 if p = /3 ǫ for arbitrary ǫ > 0. The game D 3 : b defies the probabilistic ituitio agai. However we suspect that the D 3 3 : b game agai agrees with the probabilistic

3 THE DIAMETER GAME 3 ituitio; i.e., the breakig poit should be b 0 2/3 polylog but we do ot have the courage to state a cojecture for the breakig poit for the D 3 2, b game. Theorem 3. Maker wis the game D 3 : c / l, ad Breaker wis the game D 3 : c 2, for some c, c 2 > 0, provided is big eough. Theorem 3 is implied by the followig more geeral theorem. Theorem 4. There exists a costat c 0 > 0 such that if d is a iteger, 3 d c 0 l / l l, the there is a c = c d > 0, such that Maker wis the game D d : c / l / d/2 if is large eough. Furthermore, for every a >, there is a costat c 2 > 0, depedig oly o a such that if d is a iteger, 3 d c 2 l /l l, the there exist c 3 = c 3 d > 0 ad c 4 = c 4 a, d > 0 such that Breaker wis the games D d : c 3 /d ad D d a : c 4 /d, provided is big eough. Note that i this theorem, Maker achieves diameter 2k by achievig diameter 2k for ay iteger k 2. We cojecture that the correct break poit is close up to polylog factor to the Breaker s boud. The rest of the paper is orgaized as follows: Sectio 2 describes geeral theorems ad auxiliary games, such as the degree game ad expasio game. Sectio 3 proves the result of the 2-diameter game. Sectio 4 proves the results of the d-diameter game for d Auxiliary games 2.. Positioal Games. Recall that formally a Maker-Breaker Positioal Game is defied as follows. Give a arbitrary hypergraph H = V H, EH, Maker ad Breaker take a ad b elemets of V H per tur. Maker wis by takig all elemets of at least oe edge A EH, otherwise Breaker wis. The celebrated Erdős-Selfridge theorem gives a coditio for Breaker s wi whe a = b =, see [7]. We recall its geeral, biased versio that is due to József Beck, see []. Theorem 5 Beck []. If F is the family of wiig sets of a positioal game, the Breaker has a wiig strategy i the a : b game whe + b A /a <. A F Theorem 6 Chvátal-Erdős [6]. i If F is a c-uiform family of k disjoit wiig sets, the Maker has a wiig strategy i the b : -game whe k c b i= i.

4 4 BALOGH, MARTIN, AND PLUHÁR ii If F is a c-uiform family of k disjoit wiig sets, the Maker has a wiig strategy i the b : 2-game whe c b Biased degree games. I provig theorems o the diameter games, we apply some auxiliary degree games. I such games oe player tries to distribute his moves uiformly, while the other player s goal is to obtai as may edges icidet to some vertex as possible. Give a graph G ad a prescribed degree d, Maker ad Breaker play a a : b game o the edges of G. Maker wis by gettig at least d edges icidet to each vertex. For G = K ad a = b = this game was ivestigated thoroughly i [2] ad [3]. It was show that Maker wis if d < /2 log, ad Breaker wis if d > /2 /2. This is i good agreemet with the probabilistic ituitio, sice i G,/2 the degrees of all vertices fall ito the iterval [/2 log, /2 + log ] almost surely. We are maily iterested i the case of G = K. Whe a b, aalogously oe would expect that Maker wis if d < a/a + b c log, ad Breaker wis if d > a/a + b c for some c, c > 0. Here we are iterested i givig coditios for Maker s wi oly, so this will suffice: Lemma 7. Let a /4 l ad be large eough. The Maker wis the a : b degree game o K if d < a 6ab a+b a+b l. 3/2 Proof of Lemma 7. We use a little modificatio of the weight fuctio argumet of Beck, see [3]. Cosider the hypergraph H = V H, EH, where V H = EK. The set of edges EH cotais the set A v for each vertex v K, where A v is the set of edges icidet to v. I the i th step let X i ad Y i be the set of edges selected by Maker ad Breaker, respectively. The λ, λ 2 -weight of a hyperedge A i the i th step is k w i A = + λ Y i A b/a+b+k λ 2 X i A a/a+b k, where k = 6ab a+b l ad the values of λ, λ 3/2 2 will be specified later. For ay graph edge e, let w i e = w i A ad T i = w i A. e A A We wat to esure three properties: i If Breaker wis i the i th step, the T i, ii T i+ T i, iii T 0 <. Property i is trivially true. Maker follows the greedy strategy, that is he always chooses the maximum weight edge available. Let w be the weight of the largest i= i.

5 THE DIAMETER GAME 5 weighted edge before Maker makes his last, a th, move. This meas that Maker will reduce the value of T i by at least aλ 2 w. Whe Breaker moves, he will add b edges. So, T i+ T i aλ 2 w + + λ b w. To esure Property ii, we eed to have + λ b + aλ 2. I order to esure Property iii, we require This simplifies to T 0 = + λ b/a+b k λ 2 a/a+b+k <. 2 + λ > a+b b+ka+b λ2 a ka+b b+ka+b. To satisfy ad 2, we eed a+b b+ka+b λ2 a ka+b b+ka+b < + λ + aλ 2 /b. Hece, we merely eed to verify the existece of a λ 2 > 0 that gives 3 ba+b < + aλ 2 b+ka+b λ 2 ab kba+b. Let α be the uique egative root of the equatio + x = exp{x x 2 }. Note that +x exp{x x 2 } for all x α, ad α Observe, if a+b/a 2 α 2 / l, the a/a + b 3ab a+b l, ad we have othig to prove. Otherwise oe may 3/2 substitute λ 2 = a+b l aa+ to see that 3 holds as log as 4 < α, ad use the lower boud o + x. So it is eough ba+b < exp { λ 2 ka + b 2 + λ 2 2 b a 2 a + bk aba + } ] 2ba + b l < k [a + b 2 a + b l aa + + a + b2 l aa + b a2 2b [ ] aa + l < k + b a 2 l. a + b 3/2 aa + a + b Sice it is true that b a 2 l aa + a + b a l / /2, the, by the assumptio a /4 l ad the fact that 3ab 2b aa +, iequality 4 will be satisfied if

6 6 BALOGH, MARTIN, AND PLUHÁR which ca be doe. 6ab l a + b 3/2 k, Lemma 8. Let > 2a. Breaker wis the a : b degree game o K if d > a a+b. Proof of Lemma 8. I the first roud Breaker chooses a vertex, say v, which Maker has ot touched ad chooses all of his edges to be icidet to that vertex i every roud. At the ed of the game, Maker has chose at most a a+b edges icidet to that vertex Expasio game. I the Expasio Game, Maker attempts to esure that for every pair of disjoit sets R ad S, where R = r ad S = s, there is a edge betwee R ad S. We may assume that s r. Lemma 9. Maker wis the Expasio Game with parameters r ad s if oe of the followig holds: a 2b l < r la +, b b l < r la + < 2b l ad s > c r la + < b l ad s < rb l, r la+ b l r la+. b l+r la+ Proof of Lemma 9. We use Theorem 5, Maker s ad Breaker s roles are reversed i that Theorem. The correspodig hypergraph has r r s hyperedges, each of size rs. First we prove that coditio a is sufficiet; i.e., r { + a rs/b exp s + r l rs } 5 r s b la + { exp s 2 l r } b la + <, because 2 l < r la + /b, ad we ca apply Theorem 5 directly. For b, we just plug i the appropriate value of s to 5 ad apply Theorem 5 agai. For the rest, we use similar estimates with r r s i place of r s. Hece, the sum from Theorem 5 is r + a rs/b r r s { exp s l + r b la + r } b la + < if c is satisfied here.

7 THE DIAMETER GAME 7 3. The 2-diameter game Lemma 7 immediately implies that Maker wis the D 2 a : b game if a > b are fixed ad is large eough. Actually, a eed ot be fixed, it ca be as large as /44 l /3 for large eough ad, as log as a > b, Maker wis. Maker plays a degree-game with d = 2 o K. Sice Maker ca wi this game, the graph has the property that if for vertices u ad v, where uv is ot a edge, the Nu Nv The right had side is if is odd ad 2 if is eve. This implies the diameter is Proof of Propositio. For 3 the statemet is obvious. For 4 regardless of whether Maker or Breaker starts the game, Breaker chooses a edge ot icidet to that chose by Maker. Let this edge the oe Breaker chooses be uv. The strategy of Breaker is: if Maker chooses a edge icidet to u, say uw, the Breaker chooses wv. Similarly, if Maker chooses vw, the Breaker chooses wu. Otherwise Breaker may take a arbitrary edge. Clearly, at the ed of the game, the pair {u, v} has distace at least three Proof of Theorem 2. First we prove that Breaker wis the D 2 2 : b-game if b = 2 + ǫ / l. Breaker plays i two phases. I Phase I, he picks a vertex v ad occupies all possible edges icidet to it, at least 3b/b + 2 of them. Let u,...,u t be the list of vertices so that Maker occupied the edge vu i. Note that t 3b/b b/b + 2. Now Breaker cosiders t disjoit sets of edges, for a vertex x {v, u,..., u t }, E x = {xv, xu,...,xu t }. By Theorem 6 ii, Breaker ca occupy oe of these sets, say E x, forcig v ad x at distace at least 3 from each other, if t + b 2 l t, which is satisfied for b = 2 + ǫ / l if is large eough. Now we prove that Maker wis the 2 : b game, if b is ot too large. We shall set r, s ad c such that the possible value of b is maximized. Our strategy has two phases. The first oe, which lasts 2r rouds, requires approximately 2rb + 2 edges, ad the secod deals with the rest of the 2 edges. Moreover, i the first phase, Maker will play four subgames, each with a differet strategy. Deote deg I B x to be Breaker s degree at x ad degi M x to be Maker s degree at x after Phase I. I geeral, deg B x ad deg M x will deote Breaker s ad Maker s degrees, respectively, i whichever roud the cotext idicates. Phase I. There are 2r rouds i this phase. Each of the followig games is played i successive rouds. That is, Maker plays game i i roud j iff i j mod 4. A vertex is called high if it achieves deg B x c/b before the ed of Phase I.

8 8 BALOGH, MARTIN, AND PLUHÁR The four games i Phase I will esure the followig: Game. Ratio game. If vertex x is high, the deg B x deg M x < 3b. Game 2. Degree game. For all vertices x, deg M x r. Game 3. Expasio game. If R r ad S s, the there is a Maker s edge betwee R ad S. Game 4. Coectig high vertices. We esure that every high vertex will coect to previously-high vertices by a path of legth at most 2. This subgame will cotiue i Phase II. Game verificatio. We ca view Game as a 2 : 4b-game. Here, Maker plays the degree game, for which he has a wiig strategy, provided d < / + 2b 48b 2+4b 3/2 l. This follows from Lemma 7. Suppose, at some time degb x c/b but deg M x < deg B x/3b. I that case, Breaker would have a wiig strategy for the degree game because he could take all 4b of his edges icidet to x. Eve if Maker respods by takig both edges i each roud icidet to x, we see that at the ed of this game, Breaker s degree would be at least: 6 As log as 4b deg B x + [ 3 deg B x + deg M x] [ 4b + 2 deg B x + 3 deg B x + ] 4b 3b 4b + 2 2b 3 + deg 2b + B x 6b + 3 2b c b + b 6b c 36b 3/2 l ad is large eough, we have the followig: c b6b b 3/2 l > b6b b 4b + 2 3/2 l + 3. This cotradicts the fact that Maker ca wi the degree game because it allows Breaker, from 6, to force Maker s degree of x to be less tha 48b 2b+ 4b+2 l. 3/2 Game 2 verificatio. Here, we use a greedy strategy for the 2 : 4b-degree game util the ed of Phase I i roud 2r.

9 THE DIAMETER GAME 9 Maker s strategy is to put a edge icidet to a vertex whose Maker degree is less tha r. If deg B x 3br, the Game esures that deg M x > r. If > 3b+r, which is satisfied if is large eough ad 8 4br, the it must be the case that we ca esure deg M y r for all y. Game 3 verificatio. Here we play a virtual 2 : 2 2r 2 -expasio game o sets of size r ad s. That is, for each 4b edges that Breaker chooses, Maker also assumes that Breaker also adds a set of 2 /2r 2 4b edges arbitrarily. I this way, we ca apply results about the completed game to our situatio, because Phase I is fiished i 2r rouds. This esures ot oly a Maker s edge betwee each R, S pair, but esures that it has occurred i Phase I. By Lemma 9a, it is sufficiet to have: 9 s r 3 ad 0 2 l < r l2 + l < 4r 2r2 l 3. We use /4r i place of b i the lemma. Game 4 verificatio. Let l be the maximum possible umber of high vertices. Breaker occupies 2rb edges by the ed of Phase I. Hece, c 2rb l, b implyig l 2rb2 c. For i = 0,..., 2r 4, i roud 4 il + t+4, we will work to esure that the t th 4l high vertex, deoted x t, will be coected to each of x,..., x t by a path of legth at most 2. For the games i rouds {4 il + t + 4} i 0, we play a differet : 4lb-game. Maker will choose a path of legth 2 betwee x t ad some x j, j < t. Breaker ca choose idividual edges, ad choosig a edge from a 2-path he occupies it. Maker will play as Breaker i the stadard Erdős-Selfridge game i Theorem 5. There are t wiig sets, the j th oe cosists of all of the paths of legth 2 from x j to x t which have o Breaker edges at the time that x t becomes high. Sice Game esures that, for a high vertex, x j, it is the case that deg B x j 4b deg M x j < 3b we subtract 4b from deg B x j because the effects of Game may be delayed by a roud. Hece

10 0 BALOGH, MARTIN, AND PLUHÁR deg B x j < 3b +4b < 3b +. Moreover, i the set of four rouds whe x 3b+ 3b+ 3 t becomes high, deg B x t c + 4b. As a result, the size of these wiig sets is at least b deg B x j deg B x t > 3b + c b 3 4b. If b grows slowly actually, b = O is sufficiet, this is at least b 4 c. This term is large eough to suffice, as log as c < /4. So, applyig Lemma 5, we see that Maker playig as Breaker has a wiig strategy as log as t + b /4 c/4lb <. Sice t is bouded by l 2rb2 c, the wiig strategy is sufficiet if 2rb 2 c 2 c 4c/32rb4 <. Phase II. I the odd turs of this phase, we will coect each pair of vertices that does ot curretly have a path of legth 2 betwee them. Because of Game 2 ad Game 3, for every vertex v, there are less tha s vertices that do ot have a Maker s path of legth 2 betwee them. Maker will play a game i which players will select paths of legth 2 betwee ucoected vertices. These paths do ot have a Breaker s edge so Maker gets the first move. Breaker, o the other had, chooses 2b edges. If Breaker chooses edge xy, the there are at most s paths that he occupies to coect x to other vertices. Similarly for y. As a result, this ca be viewed as a : 4bs-game betwee Maker ad Breaker. We have to compute the size of the wiig sets. If x is high ad y is ot, the whe Phase II begis, deg B x < 3b 3b ad deg B y < c b + 4b. As a result, if is large eough ad c < /4, the there are at least /4 c/b available paths of legth 2 betwee x ad y. This holds also if either x or y is high. Usig Theorem 5, we see that F s/2, Breaker playig as Maker makes 4bs moves o each term ad Maker playig as Breaker makes. Hece, Maker playig as Breaker has a wiig strategy if 2 s 2 2 4c/6b2 s <.

11 THE DIAMETER GAME We compile the iequalities which eed to be satisfied: 7, 8, 9, 0, ad 2. Let b = /8 l 8l /2, r =, s = 3/4 ad c = 2 l 8. Coditios 7, 8, 9 are trivially satisfied for these values as log as is large eough ad 2 is satisfied because s > r. Coditios 0 ad determie the proper coefficiets of b ad r. Thus, Maker wis the D 2 : b game for b = /8. 8l /2 4. Results o the geeral d-diameter game 4.. Proof of Theorem 4 Maker case. Let N i v deote the i th eighborhood of vertex v i the graph iduced by Maker s edges ad B i v = {v} i j= N jv. For i =,..., d/2 ad each vertex v, we idetify a set S i v B i v of size r i such that r 0 = ad d/2 b l r = 7 d/2 b Set r i = i r i l 2 d/2 b l + r i l2 r j, i = 2,..., d/2. β := j=0 / d/2 2 ad b := l2 l d/2 l β. Claim. There exists a positive costat c so that if d c l/ l l, the for i =,..., d/2, 6β /2 i l l 2 βi r i l l 2 βi. Proof of Claim. We will prove this by iductio o i. Let i = ad we have already defied d/2 b l r = 7 d/2 b = l l 2 β 7 l 2/β. Both bouds hold trivially.

12 2 BALOGH, MARTIN, AND PLUHÁR Let us assume the bouds hold for r i wheever i {2,..., d/2 }. Now we compute r i : r i = i r i l 2 d/2 b l + r i l 2 j=0 i r i β = + r i β r j j=0 [ ] = r i β + iβ. r i β r j Clearly, r i r i β. So, by the iductive hypothesis, r i l l2 βi. As to the lower boud, To boud this, r i r i β [ r ] i β iβ. 2/d iβ d/2 3β /2, l as log as d c l / l l for some costat c. Also r i β l 2 l2 βi l 2 l 2 β d/2 3β /2, as log as d c l for some costat c. So, we combie these ad see that r i r i β 6β /2. By the iductive hypothesis, r i 6β /2 i l l2 βi ad the proof is fiished. Maker s strategy will be to play d/2 games, playig the k th i roud l iff k l mod d/2. For i =, Maker will play the degree game. For 2 i d/2, he will play the expasio game with case c. I Game d/2, if d is odd, he plays the expasio game a o sets of size approximately l l2 β/2 d/2. If d is eve, the he will play the expasio game a to esure that B d/2 v /2.

13 THE DIAMETER GAME 3 Game. Maker plays the : d/2 b-degree game. Lemma 7 esures that Maker ca guaratee that for each v, there is a set S v B v for which S v 6 d/2 b l + d/2 b + d/2 b d/2 b l d/2 b + d/2 b 6 d/2 b l 7 = r, d/2 b as log as d/2 b / l /3. Game 2 to Game d/2. Maker plays the : d/2 b-expasio game c. Thaks to Lemma 9 c, Maker ca guaratee a set S i v B i v\ {v} i j= S jv of size S i v = = d/2 l2 d/2 l r i l 2 β l + r i l 2 i r i l 2 l 2β + r i l 2 r j = r i. j=0 i S j v Game d/2. Maker plays the : d/2 b-expasio game a. Note that, here, we have S d/2 of size r d/2 3β /2 l l2 β d/2. We just use Lemma 9 a with r = s = r d/2 i the case where d is odd ad s = + /2 i the case where d is eve. Of course, a = ad we use d/2 b istead of b. To verify that the coditios of the lemma are satisfied: j= 2 l 2β < 3β /2 d/2 l β d/2 < r d/2 l 2 ad this directly satisfies the coditio i Lemma 9 a Proof of Theorem 4 Breaker case, a =. We will show that Breaker ca wi if a =, b 4d /d /d ad is sufficietly large. I the first move, Breaker will choose a edge betwee u ad v. If Maker goes first, we make sure that either u or v is icidet to Maker s first edge. At each move, Breaker will play two roles. He will use d /d /d edges to esure the maximum Maker degree is small ad the will use the remaiig edges to esure that every vertex i the i th Maker s eighborhood of u is adjacet by a Breaker edge to every vertex i the j th Maker s eighborhood of v for j = 0,...,d i.

14 4 BALOGH, MARTIN, AND PLUHÁR If Breaker devotes b = d /d /d edges to the maximum-degree game, Lemma 7 Maker ad Breaker switch roles ad d is replaced with d says that Breaker has a wiig strategy to keep the maximum degree,, at most 3 b + + 6b l b + 3/2 l + 6 b b /d + 6d /2d 2 d /d l { } /d l exp 6d /2d 2. d /d The umber of edges used ext is b 2. We will guaratee that, for all k {0,...,d}, B k u B d k v =. Let xy be the edge chose by Maker i the precedig move. First, we will cosider the distace from u: i Let x be at least as close as y to u; i.e., let i = distx, u disty, u, where dist is the legth of the shortest path i the graph iduced by Maker s edges. If i d 2, the Breaker will choose each edge i N k y, d i 2 k l=0 N l v, for k = 0,...,d i 2; ad each edge i N k y, i k l=0 N lu, for k = 0,...,i. ii Let j = mi{distx, v, disty, v}. There are two possibilities: If j = disty, v distx, v, the Breaker will choose each edge i N k x, d j 2 k l=0 N l u, for k = 0,..., d j 2; ad each edge i N k x, j k l=0 N lv, for k = 0,...,j. Otherwise, let j = distx, v < disty, v, the Breaker will choose each edge i N k y, d j 2 k l=0 N l u, for k = 0,..., d j 2; ad each edge i N k y, j k l=0 N lv, for k = 0,..., j. This procedure will esure that if there is some edge betwee N i u ad N j v that i + j > d.

15 THE DIAMETER GAME 5 We will, without loss of geerality, focus o item i. The total umber of edges eeded is at most: 4 d i 2 k=0 = = d i 2 k k l=0 d i 2 k=0 i l + k=0 i k k l=0 l d i k + i i k k=0 [ d i d i d i d i + ] 2 [ + i i+ i + i + ]. 2 Claim 2. Let, d be positive itegers. The for i = 0,...,d, fi := [ d i d i d i d i + ] + [ i i+ i + i + ] d d d d +. Proof of Claim 2. By symmetry, fi = fd i. So it is sufficiet to show that fi fi + for i {0,..., d 3/2 }. fi fi + = [ d i d i d i d i + ] + [ i i+ i + i + ] [ d i 2 d i d i d i 2 + ] [ i + i+2 i + 2 i+ + ] = [ d i 2 d i 2 2d 2i 2 + d i ] [ i i + 2 2i i + ] = 2 i d i Sice i d 3/2, [ d 2i+ d 2i+ > ] i +. d i + i + d i +. Thus, it must be the case that fi attais its maximum at i = 0 or i = d ad the claim is prove. We boud the term 4 as follows: [ d d d d + ] { } 2 d d 2 exp. 2

16 6 BALOGH, MARTIN, AND PLUHÁR This umber of edges is sufficiet to guaratee i ad by symmetry is eough to guaratee either coditio of ii. We boud the sum as follows: { } { 2 } d 2 2d d 2 d exp 2d exp 6d d 2d l d 2d d /d { } 2d /d /d exp 3d 2d 2d 2 l. /d As log as d l ad is large eough, we have that 2ll { } 2d /d /d exp 3d 2d 2d 2 l 3d /d /d. /d Hece, we oly eed b + b 2 4d /d /d i order to esure a Breaker wi Proof of Theorem 4 Breaker case, a 2. For the sake of simplicity, we deal oly with the case a = 2; the geeral case is very similar. We will show that Breaker ca wi if b 4 /d ad will do so by simply playig the degree game. As we have see i 3, Breaker ca esure that 2 l b + 2 b l /d 4 /d /d /d l 2/3 /d, as log as is large eough. With beig the maximum degree, for ay vertex v, as log as 4. d B d v + i i=0 = + d d < 2 d, 2

17 THE DIAMETER GAME 7 But the, usig the upper boud for ad recallig d 2, d 2 B d v < 2 d < 2 3 /d 8 9. So, for ay vertex v, there is at least oe vertex of distace greater tha d from it. Note that i order for Breaker to wi this degree game, we eed to have that b /4 l. Sice b = 4 /d, this holds for d l /2 ll ad large eough. 5. Thaks We thak Agelika Steger for ucoverig a flaw i a previous versio of the mauscript ad József Beck for reviewig it ad makig helpful commets. Refereces [] J. Beck, Remarks o positioal games. Acta Math Acad Sci Hugar , [2] J. Beck, Radom graphs ad positioal games o the complete graph. Radom graphs 83 Pozań, 983, 7 3, North-Hollad Math. Stud., 8, North-Hollad, Amsterdam, 985. [3] J. Beck, Determiistic graph games ad a probabilistic ituitio. Combiatorics, Probability ad Computig 3 994, [4] J. Beck, Positioal Games. Combiatorics, Probability ad Computig , [5] M. Bedarska, ad T. Luczak, Biased positioal games ad the phase trasitio. Radom Structures ad Algorithms 8 200, [6] V. Chvátal ad P. Erdős, Biased positioal games. Algorithmic aspects of combiatorics Cof., Vacouver Islad, B.C., 976. A. Discrete Mathematics 2 978, [7] P. Erdős ad J. L. Selfridge, O a combiatorial game. J. Combiatorial Theory Series A 4 973, [8] A. Pluhár, Geeralized Harary Games. Acta Cyberetica [9] A. Pluhár, The accelerated k-i-a-row game. Theoretical Computer Sciece , [0] N. Siebe, Hexagoal polyomio weak, 2-achievemet games. Acta Cyberetica , [] T. Szabó ad M. Stojaković, Positioal games o radom graphs. Radom Structures ad Algorithms , o. -2, [2] L. A. Székely, O two cocepts of discrepacy i a class of combiatorial games. Fiite ad Ifiite Sets, Colloq. Math. Soc. Jáos Bolyai, Vol. 37, North-Hollad, 984, Departmet of Mathematical Scieces, Uiversity of Illiois address: jobal@math.uiuc.edu Departmet of Mathematics, Iowa State Uiversity address: rymarti@iastate.edu Uiversity of Szeged, Departmet of Computer Sciece address: pluhar@if.u-szeged.hu