Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite

Abstract

Erdős and Simonovits asked the following question: For an integer $r\ge 2$ and a family of non-bipartite graphs $H$, determine the infimum of $\alpha$ such that any $H$-free $n$-vertex graph with minimum degree at least $\alpha n$ has chromatic number at most $r$. We answer this question for $r=2$ and any family consisting of odd cycles. Let $C$ be a family of odd cycles in which $C_{2\ell +1}$ is the shortest odd cycle not in $C$ and $C_{2k+1}$ is the longest odd cycle in $C$, we show that if $G$ is an $n$-vertex $C$-free graph with $n\ge 1000k^8$ and $\delta \left(G\right)>max\{n/\left(2(2\ell +1)\right),2n/(2k+3)\}$, then $G$ is bipartite. Moreover, this bound on the minimum degree is tight.