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arXiv 1210.4342math.CO
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有偏奇循环博弈

The biased odd cycle game

Asaf Ferber, Roman Glebov, Michael Krivelevich, Hong Liu, Cory Palmer, Tomas Valla, Mate Vizer

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10 pages
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摘要：

在本文中，我们考虑有偏见的Maker-Breaker游戏上的边集的一个给定的图$G$。我们证明了对于每一个$δ>0$和足够大的$n$，存在一个常数$k$，如果$δ(G)\geq δn$和$χ(G)\geq k$，那么Maker可以在$(1:b)$博弈中为$b=O(\frac{n}{\log^2 n})$建立一个奇循环.我们也考虑类似的游戏，其中制造商和断路器要求顶点，而不是边缘。这是一个特殊的情况下，以下众所周知的和臭名昭著的困难的问题，由于Duffus，Zuuczak和Rödl：是真的，对于任何正常数$t$和$b$，存在一个整数$k$，使每一个图$G$，如果$χ(G)\geq k$，然后制造商可以建立一个图，这是不$t$ -着色，在$(1:b)$ Maker-Breaker游戏玩的顶点$G$？

英文摘要：

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $δ>0$ and large enough $n$, there exists a constant $k$ for which if $δ(G)\geq δn$ and $χ(G)\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O(\frac{n}{\log^2 n})$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, Łuczak and Rödl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $χ(G)\geq k$, then Maker can build a graph which is not $t$-colorable, in the $(1:b)$ Maker-Breaker game played on the vertices of $G$?

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