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arXiv · 1210.4342

The biased odd cycle game

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

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $\delta>0$ and large enough $n$, there exists a constant $k$ for which if $\delta(G)\geq \delta n$ and $\chi(G)\geq k$, 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, {\L}uczak and R\"{o}dl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $\chi(G)\geq k$, 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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Asaf Ferber, Roman Glebov, Michael Krivelevich, Hong Liu, Cory Palmer, Tomas Valla, Mate Vizer. 2012-10-16. The biased odd cycle game. https://arxiv.org/abs/1210.4342

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