A BOUND FOR THE COPS AND ROBBERS PROBLEM *

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1 SIAM J. DISCRETE MATH. Vol. 25, No. 3, Society for Industrial and Alied Mathematics A BOUND FOR THE COPS AND ROBBERS PROBLEM * ALEX SCOTT AND BENNY SUDAKOV Abstract. In this short aer we study the game of cos and robbers, which is layed on the vertices of some fixed grah G. Cos and a robber are allowed to move along the edges of G, and the goal of cos is to cature the robber. The co number cðgþ of G is the minimum number of cos required to win the game. ffiffiffi Meyniel conjectured a long time ago that Oð n Þ cos are enough for any connected G on n vertices. Imroving ffiffiffiffiffiffiffi several revious results, we rove that the co number of an n-vertex grah is at most n2 ð1þoð1þþ.a similar result indeendently and slightly before us was also obtained by Lu and Peng. Key words. ursuit-evasion games, grah searching, co number, games on grahs AMS subject classifications. Primary, 05C57; Secondary, 91A24, 91A43 DOI / Introduction. Let G be a simle, undirected, connected grah on n vertices. The game of Cos and Robbers, which was introduced almost thirty years by Nowakowski and Winkler [12] and by Quilliot [13], is layed on the vertices of G as follows. There are two layers: a set of k 1 cos and one robber. The game begins by the cos occuying some set of k vertices of G and then the robber also chooses a vertex. Afterward they move alternatively, first cos and then the robber, along the edges of grah G. At every ste each co or robber is allowed to move to any neighboring vertex or do nothing and stay where they are. Multile cos are allowed to occuy the same vertex. The cos win if at some time, there is a co at the same vertex as the robber; otherwise, the robber wins. The minimum number of cos for which there is a winning strategy, no matter how robber lays, is called the co number of G and is denoted by cðgþ. Note that this number is clearly at most n and also that the initial osition of cos does not matter, since G is connected. The co number was introduced by Aigner and Fromme [1] who roved, for examle, that if G is lanar, then cðgþ 3. They also observed that if G has girth at least 5 (i.e., no cycles of length 4), then its co number is at least the minimum degree of G.In articular, this together with the well-known construction of dense grahs of girth 5 ffiffiffi shows that there are n-vertex grahs which require at least Ωð n Þ cos. A quarter century ago, Meyniel conjectured that this is tight and that Oð n Þ cos are always suffi- ffiffiffi cient. The first nontrivial uer bound for this roblem was obtained by Frankl [8], who n log roved that cðgþ Oð Þ. This was later imroved by Chiniforooshan [7] to Oð n Þ. All logarithms in this aer are binary. The game of cos and robbers was also studied by Andreae [4], Berarducci and Intrigila [5], and Alsach [3] and for random grah by Bollobás, Kun, and Leader [6] and by Łuczak and Prałat [11]. The aim of this short aer is to rove the following new bound for the cos and robber roblem. *Received by the editors October 27, 2010; acceted for ublication (in revised form) July 1, 2011; ublished electronically Setember 20, htt:// Mathematical Institute, St Giles, Oxford, OX1 3LB, UK (scott@maths.ox.ac.uk). Deartment of Mathematics, University of California, Los Angeles, Los Angeles, CA (bsudakov@ math.ucla.edu). This author s research was suorted in art by NSF grant DMS , NSF CAREER award DMS , and by USA Israeli BSF grant

2 THEOREM n2 ð1þoð1þþ A BOUND FOR THE COPS AND ROBBERS PROBLEM The co number of any connected n-vertex grah is at most. 2. Proof of the main result. The roof of the main theorem has two ingredients. The first is a result of Aigner and Fromme which says that one co can control the shortest ath between two vertices. Let P be the shortest ath in G between two vertices u and v. Then the following lemma was roved in [1]. LEMMA 2.1. One co can move along the vertices of P such that, after a finite number of stes, if the robber ever visits P, then he will be caught in the next ste. This result can be used for grahs with large diameter, since we can delete a long ath from such a grah and use induction. The case where G has relatively small diameter will be treated using the following key lemma, which we think has indeendent interest. LEMMA 2.2. Let G be a grah on n 2 30 vertices with diameter D 2 log 3 n. Then cðgþ nðþ 3 2. Proof. Throughout the roof, we assume that n is large enough for our estimates to hold. Let t ¼ ffiffiffiffiffiffiffiffiffiffiffi 3 log. Consider random subsets C 1 ; :::;C tþ1 of G. For every vertex of G we ut it into each C j with robability ðþ 2 2 (so a single vertex may be in many C i ). Since jc j j is binomially distributed with exectation μ ¼ nðþ 2 2, by the standard Chernoff-tye estimates (see, e.g., Aendix A in [2]), we have that the robability that C j has more than 2μ ¼ 2nðÞ 2 2 vertices is at most e μ 3 < n 2. Then, with robability at least 1 > 0.9, the n 2 sum of the sizes of these sets is at most nðþ 3 2, which will give our final bound. For every subset A of G and integer i let BðA; iþ be the ball of radius i around A, i.e., all the vertices of G which can be reached from some vertex in A by a ath of length at most i. We need the following simle claim. Claim 2.3. The following statement holds with robability 0.9: for every A V ðgþ such that jaj n2, every i t such that jbða; 2 i Þj 2 jaj, and every j, jbða; 2 i Þ C j j jaj: Proof. Let jaj ¼a. Note that for any fixed A, i, j the number of oints from C j in BðA; 2 i Þ is binomially distributed with exectation at least a log 2 n. Thus by Chernofftye estimates, the robability that it is smaller than a is at most e a log2 n 3. The number of sets of size a is ð n aþ and the number of airs of indices i, j is at most, and the result follows since P að n a Þe a log2 n 3 < 0.1. Thus we can choose subsets C 1 ; :::;C tþ1 that satisfy the assertion of this claim and such that the sum of their sizes is at most nðþ 3 2. We lace at each vertex u of G one co for each set C j that contains u. We will show that these cos can always catch the robber. Suose that the robber is located at vertex v of the grah G. Since the cos move first, note that the degree of this vertex is at most 2. Otherwise, by the claim, we already have a co in its neighborhood who will catch the robber in the first move. Consider the following sequence of sets A i, D i for i ¼ 1; :::tþ 1 defined recursively. Let A 1 be the largest subset of Bðv; 1Þ such that jbða 1 ; 1Þj < 2 ja 1 j, and let D 1 ¼ Bðv; 1Þ A 1. Note that for any X D 1 we have that jbðx;1þj 2 jxj since

3 1440 ALEX SCOTT AND BENNY SUDAKOV otherwise we could increase A 1 by adding X to it. Thus by Claim 2.3 we have that BðX;1Þ contains more than jxj oints from C 1. Consider an auxiliary biartite grah with vertex classes D 1 and C 1, where we join a vertex u in D 1 to w in C 1 if w is within distance at most one from u. Note that, by the above discussion, it follows from Hall s theorem that this biartite grah has a comlete matching from D 1 to C 1. Therefore for every vertex u D 1 we can choose a corresonding vertex w in C 1, which is within distance one from u and such that all w are distinct. We can therefore ask the cos in C 1 to occuy D 1 in the first round. Now suose we have already defined A i, D i. Consider the ball BðA i ; 2 i 1 Þ. Let A iþ1 be the largest subset of BðA i ; 2 i 1 Þ such that jbða iþ1 ; 2 i Þj < 2 ja iþ1 j, and let D iþ1 ¼ BðA i ; 2 i 1 Þ A iþ1. Again for any subset X D iþ1 we have that jbðx;2 i Þj 2 jxj. Thus by Claim 2.3 we have that BðX;2 i Þ contains more than jxj oints from C iþ1. As above, we consider an auxiliary biartite grah with arts D iþ1 and C iþ1 such that a vertex u in D iþ1 is joined to w in C iþ1 if w is within distance at most 2 i from u. The biartite grah has a comlete matching from D iþ1 to C iþ1, and so for every vertex u D iþ1 we can choose a corresonding vertex w in C iþ1, which is within distance at most 2 i from u and such that all w are distinct. We will ask the cos in C iþ1 to occuy D iþ1 by the end of the first 2 i rounds (by rounds we mean here that cos will move first, then the robber moves, and so on until both the cos and robber have each made 2 i moves). Next we claim that there is an index s t þ 1 such that A s is emty. Indeed note that by construction we have ja iþ1 j jbða i ; 2 i 1 Þj 2 ja i j. Therefore if A tþ1 is not emty, we have that ja tþ1 j ð2 Þ tþ1 n2 2 : Moreover, we also have that BðA tþ1 ; 2 t Þ has size at most 2 ja tþ1 j n, and this contradicts the assumtion that the diameter of G is at most 2 log 3 n. Let s t þ 1 be such that A s is emty. We have shown that we can move cos from the sets C i, 1 i s, such that after 2 i 1 rounds they will occuy the set D i. Also we have that BðA s 1 ; 2 s 2 Þ¼D s. Now we claim that after 2 s 1 rounds we have already caught the robber. After one round the robber has made one ste, so he is in Bðv; 1Þ. Note that D 1 is already occuied by cos after the first round, so in order to survive the robber must be in A 1 ¼ Bðv; 1Þ D 1. Suose by induction that after 2 i 1 rounds the robber is in the set A i. Consider the next 2 i 1 rounds (so in total 2 2 i 1 ¼ 2 i rounds). After this many additional rounds, the robber is in BðA i ; 2 i 1 Þ. Since we know that D iþ1 is occuied by the cos by time 2 i, the robber must be in BðA i ; 2 i 1 Þ D iþ1 ¼ A iþ1. Thus, arguing by induction, the only lace the robber can be after 2 i 1 rounds without having been caught is in the set A i. Since A s is emty, we are done. This comletes the roof of the lemma. Proof of Theorem 1.1. Let G be a connected grah on n vertices. We rove by induction that cðgþ fðnþ ¼2nðÞ 3 2. Since cðgþ n, the result holds trivi- ally when 2ðÞ Thus we can assume that 400. If the diameter of G is at most 2 log 3 n, then we are immediately done by Lemma 2.2. Otherwise G contains two vertices such that the shortest ath P between them has length at least D ¼ 2 log 3 n. Put one co on this ath. By Lemma 2.1, after a finite number of

4 A BOUND FOR THE COPS AND ROBBERS PROBLEM 1441 stes this co can revent the robber from entering ath P. Thus we can continue the game on the grah obtained from G by deleting P. If this grah is disconnected, we continue laying on the connected comonent which contains the robber. Thus cðgþ 1 þ cðg PÞ. By induction we know that ð1þ cðg PÞ fðn DÞ 2ðn DÞðÞ 3 2 ffiffiffiffiffiffi logðn DÞ 2nðÞ 3 2 ffiffiffiffiffiffi logðn DÞ 2: Since 400,wehaveD 2 20 n. Then logðn DÞ 2D n, and therefore ffiffiffiffiffiffiffiffiffiffiffiffiffiffi logðn DÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D n ffiffiffiffiffiffiffiffiffiffiffi 2D n ffiffiffiffiffiffiffiffiffiffiffi : 2D ðn We also have that 2 Þ ð1 þ 2D ðn ffiffiffi ÞÞ. Substituting this into (1) we get that ffiffiffiffiffiffi cðg PÞ 2nðÞ 3 2 logðn DÞ 2 2nðÞ 3 2 þ2d ðn Þ 2 2nðÞ þ 2D n ffiffiffiffiffiffiffiffiffiffiffi 2 ¼ fðnþþffiffiffi 4 2 < fðnþ 1: Therefore cðgþ 1 þ cðg PÞ < 1 þ fðnþ 1 ¼ fðnþ, which comletes the induction ste. Finally, let us note that the strategy in Lemma 2.2 does not require the cos to see the robber s movements after the first round. The cos therefore have the following randomized strategy against an invisible robber (see [9] for a discussion of cos and robbers games where there is a co with limited vision). The cos move to their assigned ositions, then guess the osition of the robber and follow the strategy of Lemma 2.2, reeating this rocess until the robber is caught. This algorithm catches the robber in olynomial exected time. On a general grah, we can add a further co on each ath P that we have deleted in the roof of Theorem 1.1: If these cos move at random on their aths, the robber will be caught in finite (although not necessarily olynomial) exected time. Note added in roof. When this aer was written, we learned that a similar result indeendently and slightly before us was also obtained by Lu and Peng [10]. We would like to thank Krivelevich for bringing their aer to our attention. REFERENCES [1] M. AIGNER AND M. FROMME, A game of cos and robbers, Discrete Al. Math., 8 (1984), [2] N. ALON AND J. SPENCER, The Probabilistic Method, 3rd ed., Wiley, Hoboken, NJ, [3] B. ALSPACH, Searching and sweeing grahs: A brief survey, Matematiche (Catania), 59 (2004), [4] T. ANDREAE, On a ursuit game layed on grahs for which a minor is excluded, J. Combin. Theory Ser. B, 41 (1986), [5] A. BERARDUCCI AND B. INTRIGILA, On the co number of a grah, Adv. in Al. Math., 14 (1993), [6] B. BOLLOBÁS, G. KUN, AND I. LEADER, Cos and Robbers in Random Grahs, ArXiv , [7] E. CHINIFOROOSHAN, A better bound for the co number of general grahs, J. Grah Theory, 58 (2008),

5 1442 ALEX SCOTT AND BENNY SUDAKOV [8] P. FRANKL, Cos and robbers in grahs with large girth and Cayley grahs, Discrete Al. Math., 17 (1987), [9] V. ISLER AND N. KARNAD, The role of information in the co-robber game, Theoret. Comut. Sci., 399 (2008), [10] L. LU AND X. PENG, On Meyniel s conjecture of the co number, J. Grah Theory, to aear. [11] T. ŁUCZAK AND P. PRAŁAT, Chasing robbers on random grahs: Zigzag theorem, Random Structures Algorithms, 37 (2010), [12] R. NOWAKOWSKI AND P. WINKLER, Vertex-to-vertex ursuit in a grah, Discrete Math., 43 (1983), [13] A. QUILLIOT, Jeux et ointes fixes sur les grahes, Ph.D. dissertation, Université de Paris VI, Paris, 1978.