Minimum degree stability of C 2 k + 1 ${C}_{2k+1}$ -free graphs

We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a $C_{2k+1}$-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a $\{C_3,C_5,\text{…},C_{2k+1}\}$-free graph on $n$ vertices has minimum degree greater than $\frac{2}{2k+3}n$, then it is bipartite. Häggkvist showed that for $k\in \left\{1,2,3,4\right\}$, if a $C_{2k+1}$-free graph on $n$ vertices has minimum degree greater than $\frac{2}{2k+3}n$, then it is bipartite. Häggkvist also pointed out that this result cannot be extended to $k\ge 5$. In this paper, we give a complete answer for any $k\ge 5$. We show that if $k\ge 5$ and $G$ is an $n$-vertex $C_{2k+1}$-free graph with $\delta \left(G\right)\ge \frac{n}{6}+1$, then $G$ is bipartite, and the bound $\frac{n}{6}+1$ is tight.