Hitting time results for Maker‐Breaker games
Hitting time results for Maker‐Breaker games
复制标题
Maker-Breaker 游戏的击球时间结果
DOI：
10.1002/rsa.20392
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发表时间：
2010
影响因子：
1
通讯作者：
Michael Krivelevich
中科院分区：
文献类型：
--
作者：
Sonny Ben;Asaf Ferber;Dan Hefetz;Michael Krivelevich
我们研究在随机图的边缘上玩的制造商 - 突破性游戏。 \ documentClass {article} \ usepackage {Mathrsfs} \ usepackage {amsmath，amssymb} \ pagestyle {empty} \ begin {document} \ begin {document} \ begin {align*} \ mathcal {align*} \ mathcal {p} p} {p} {align {align {align {align {align {align*}我们专注于制造商图的三个天然目标属性，即与k -vertex相互连接，承认完美的匹配，并成为哈密顿人。随机图最小程度2K，具有高概率制造商在随机图中赢得了完美的匹配游戏，首先以高概率达到最小程度制造商在随机图中赢得了汉密尔顿游戏，而后者的两个陈述结算了Stojaković和Szabó的猜想。随机图
We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011