An improved result on the stability of odd cycles

Let $C$ be a family consisting of some odd cycles. Suppose that $C_{2\ell +1}$ is the shortest odd cycle not in $C$ and $C_{2k+1}$ is the longest odd cycle in $C$. Let $B{C}_{2\ell +1}\left(n\right)$ denote the graph obtained by taking $2\ell +1$ vertex-disjoint copies of $K_{\frac{n}{2(2\ell +1)},\frac{n}{2(2\ell +1)}}$ and selecting a vertex in each of them such that these vertices form a cycle of length $2\ell +1$. In this paper, we show that if $k\ge {79}^4{\ell }^{12}$, $n\ge 4(2\ell +1)k+(16\ell +10)(2\ell +1)k^{\frac{3}{4}}$ and G is an n-vertex $C$-free graph with minimum degree $\delta \left(G\right)>\frac{n}{2(2\ell +1)}$, then G is bipartite. The condition on the minimum degree is tight evidenced by $B{C}_{2\ell +1}\left(n\right)$. Furthermore, we show the only non-bipartite $C$-free graph with minimum degree $\frac{n}{2(2\ell +1)}$ is $B{C}_{2\ell +1}\left(n\right)$. This improves the condition of n in a result of Yuan-Peng. The previous known result of Yuan-Peng corresponding to the case $C=\left\{C_{2k+1}\right\}$ for $k\ge 5$ (thus $\ell =1$) implies that if G is an n-vertex non-bipartite graph with $\delta \left(G\right)\ge \frac{n}{6}$ and $G\ne B{C}_3\left(n\right)$, then $C_{2k+1}\subset G$ for all $k\in [5,\frac{n}{21000}]$. Taking $C=\left\{C_{2k+1}\right\}$ in our main result, with the improvement of the requirement on n in this paper, we can extend the range of k from $[5,\frac{n}{21000}]$ to $[5,(\frac{1}{12}-o\left(1\right)\left)n\right]$. Precisely, our main result implies that if G is an n-vertex non-bipartite graph with $\delta \left(G\right)\ge \frac{n}{6}$ and $G\ne B{C}_3\left(n\right)$, then $C_{2k+1}\subset G$ for all $k\in [5,(\frac{1}{12}-o\left(1\right)\left)n\right]$. Furthermore, we give an example to illustrate the condition on n given here is asymptotically tight.