  Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq
2$ and a family of non-bipartite graphs $\mathcal{H}$, what is the tight bound
of $\alpha$ such that any $\mathcal{H}$-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 of odd cycles. Let $l\le k$ and $n\ge
1000k^{8}$ be positive integers. Let ${\mathcal C}$ be a family of odd cycles,
$C_{2l+1}$ be the shortest odd cycle not in ${\mathcal C}$, and $C_{2k+1}$ be
the longest odd cycle in ${\mathcal C}$. Let $BC_{2l+1}(n)$ denote the graph
obtained by taking $2l+1$ vertex-disjoint copies of
$K_{\frac{n}{2(2l+1)},\frac{n}{2(2l+1)}}$ and selecting a vertex in each of
them such that these vertices form a cycle of length $2l+1$. Let $C_{2k+3}({n
\over 2k+3})$ denote the balanced blow up of $C_{2k+3}$ with $n$ vertices. Note
that both $BC_{2l+1}(n)$ and $C_{2k+3}({n \over 2k+3})$ are $n$-vertex
${\mathcal C}$-free non-bipartite graphs. We show that if $G$ is an $n$-vertex
${\mathcal C}$-free graph with $\delta(G)>\max\{ \frac{n}{2(2l+1)},
\frac{2}{2k+3}n\}$, then $G$ is bipartite. The bound is tight evident by
$BC_{2l+1}(n)$ and $C_{2k+3}({n \over 2k+3})$. Moreover, the only $n$-vertex
${\mathcal C}$-free non-bipartite graph with minimum degree $\max\{
\frac{n}{2(2l+1)}, \frac{2}{2k+3}n\}=\frac{n}{2(2l+1)}$ is $BC_{2l+1}(n)$, and
the the only $n$-vertex ${\mathcal C}$-free non-bipartite graph with minimum
degree $\max\{ \frac{n}{2(2l+1)}, \frac{2}{2k+3}n\}=\frac{2}{2k+3}n$ is
$C_{2k+3}({n \over 2k+3})$. Our result also unifies stability results of
Andr\'{a}sfai, Erd\H{o}s and S\'{o}s, H\"{a}ggkvist and Yuan and Peng for large
$n$.
