On the structure of dense graphs with given odd girth

A classical theorem of Andr\'asfai, Erd\H{o}s, and S\'os states that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\delta(G)>\frac{2n}{2k+1}$ is bipartite (i.e., homomorphic to $K_2$). Messuti and Schacht proved that the same odd girth condition with $\delta(G)>\frac{3n}{4k}$ forces a homomorphism to $C_{2k+1}$. In this paper, we strengthen the above results by showing that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\delta(G)>\frac{4n}{6k-1}$ is homomorphic to the M\"obius ladder on $4k$ vertices. This answers a question of Messuti and Schacht and generalizes a result of Brandt and Ribe-Baumann.