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A threshold for the Maker‐Breaker clique game
Author(s) -
Müller Tobias,
Stojaković Miloš
Publication year - 2014
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20489
Subject(s) - combinatorics , mathematics , graph , discrete mathematics , conjecture , random graph , random regular graph , mathematical economics , complete graph , clique , circuit breaker , stochastic game , chordal graph , 1 planar graph , physics , quantum mechanics
ABSTRACT We study the Maker‐Breaker k ‐clique game played on the edge set of the random graph G ( n, p ). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G ( n, p ), until all the edges are claimed. Maker wins if he claims all the edges of a k ‐clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is atn − 2 k + 1, for all k > 3, thus proving a conjecture from Ref. [Stojaković and Szabó, Random Struct Algor 26 (2005), 204–223]. More precisely, we conclude that there exist constants c , C > 0 such that when p > C n − 2 k + 1the game is Maker's win a.a.s., and when p < c n − 2 k + 1it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K 5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G ( n, p ). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 318–341, 2014