  A graph $G$ is called $C_{2k+1}$-free if it does not contain any cycle of
length $2k+1$. In 1981, Haggkvist, Faudree and Schelp showed that every
$n$-vertex triangle-free graph with more than $\frac{(n-1)^2}{4}+1$ edges is
bipartite. In this paper, we extend their result and show that for $1\leq t\leq
2k-2$ and $n\geq 318t^2k$, every $n$-vertex $C_{2k+1}$-free graph with more
than $\frac{(n-t-1)^2}{4}+\binom{t+2}{2}$ edges can be made bipartite by either
deleting at most $t-1$ vertices or deleting at most
$\binom{\lfloor\frac{t+2}{2}\rfloor}{2}+\binom{\lceil\frac{t+2}{2}\rceil}{2}-1$
edges. The construction shows that this is best possible.

Combinatorics