The biased odd cycle game

TL;DR

This paper investigates biased Maker-Breaker games on graphs, showing Maker can create an odd cycle under certain conditions, and explores a vertex-based variant related to a longstanding open problem in graph theory.

Contribution

It proves Maker can form an odd cycle in biased games on graphs with high minimum degree and chromatic number, and examines a vertex-based version linked to a famous open problem.

Findings

Maker can build an odd cycle in certain biased games on dense graphs.

The vertex-based game variant relates to a major open problem in graph coloring.

Results depend on graph density and chromatic number thresholds.

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 $G$?