arXiv 1210.76182012-10-29math.CO

随机棋盘上的偏见游戏

Biased Games On Random Boards

Asaf Ferber, Roman Glebov, Michael Krivelevich, Alon Naor

摘要：

在本文中，我们分析了有偏见的制造商-断路器游戏和避免者-执行者游戏，都发挥了随机板$G\sim \gnp$的边缘集。在制造者-破坏者游戏中，有两个玩家，分别被称为制造者和破坏者。在每一轮中，Maker声明$G$的一个先前未声明的边缘，而Breaker则通过声明$b$先前未声明的边缘来响应。我们考虑的Hamilton游戏，完美匹配游戏和$k$ -顶点连通性游戏，其中制造商的目标是建立一个图，拥有相关的属性。避免者-执行者博弈是制造者-破坏者博弈的反向模拟，略有修改，其中两个玩家每次移动时分别声明至少1条和至少$b$先前未声明的边，而避免者旨在避免构建具有相关属性的图。Maker-Breaker游戏被称为“偏置单调”，也就是说，如果Maker赢得了$(1,b)$游戏，他也赢得了$(1,b-1)$游戏。因此，将一个游戏的关键偏差$b^*$定义为游戏的“断点”是有意义的。也就是说，每当$b\leq b^*$出现时，Maker都会赢得$(1,b)$游戏，否则就会失败。关键偏差的类似定义存在于回避者-实施者游戏中：在这里，游戏$b^*$的关键偏差是这样的：对于每一个$b > b^*$，回避者都赢得$(1,b)$游戏，否则就输了。我们证明，对于每一个$p=ω(\frac{\ln n}{n})$，$G\sim\gnp$是典型的，所有上述庄家-断路器游戏的临界偏差是渐近$b^*=\frac{np}{\ln n}$。我们还证明了在$p=Θ(\frac{\ln n}{n})$的情况下，临界偏差是$b^*=Θ(\frac{np}{\ln n})$。这些结果解决了Stojaković和Szabó的一个猜想。对于避免者-执行者博弈，我们证明了对于$p=Ω(\frac{\ln n}{n})$，所有上述博弈的关键偏差都是$b^*=Θ(\frac{np}{\ln n})$。

英文摘要：

In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on the edge set of a random board $G\sim \gnp$. In Maker-Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of $G$ and Breaker responds by claiming $b$ previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the $k$-vertex-connectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider-Enforcer games are the reverse analogue of Maker-Breaker games with a slight modification, where the two players claim at least 1 and at least $b$ previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property. Maker-Breaker games are known to be "bias-monotone", that is, if Maker wins the $(1,b)$ game, he also wins the $(1,b-1)$ game. Therefore, it makes sense to define the critical bias of a game, $b^*$, to be the "breaking point" of the game. That is, Maker wins the $(1,b)$ game whenever $b\leq b^*$ and loses otherwise. An analogous definition of the critical bias exists for Avoider-Enforcer games: here, the critical bias of a game $b^*$ is such that Avoider wins the $(1,b)$ game for every $b > b^*$, and loses otherwise. We prove that, for every $p=ω(\frac{\ln n}{n})$, $G\sim\gnp$ is typically such that the critical bias for all the aforementioned Maker-Breaker games is asymptotically $b^*=\frac{np}{\ln n}$. We also prove that in the case $p=Θ(\frac{\ln n}{n})$, the critical bias is $b^*=Θ(\frac{np}{\ln n})$. These results settle a conjecture of Stojaković and Szabó. For Avoider-Enforcer games, we prove that for $p=Ω(\frac{\ln n}{n})$, the critical bias for all the aforementioned games is $b^*=Θ(\frac{np}{\ln n})$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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