  We study minimum degree conditions for which a graph with given odd girth has
a simple structure. For example, the classical work of Andr\'asfai, Erd\H os,
and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum
degree bigger than $\frac{2}{2k+1}n$ must be bipartite. We consider graphs with
a weaker condition on the minimum degree. Generalizing results of H\"aggkvist
and of H\"aggkvist and Jin for the cases $k=2$ and $3$, we show that every
$n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than
$\frac{3}{4k}n$ is homomorphic to the cycle of length $2k+1$. This is best
possible in the sense that there are graphs with minimum degree $\frac{3}{4k}n$
and odd girth $2k+1$ which are not homomorphic to the cycle of length $2k+1$.
Similar results were obtained by Brandt and Ribe-Baumann.
