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We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erdős, and Sós implies that every  n ‐vertex graph with odd girth    2  k  +  1    and minimum degree bigger than2   2  k  +  1    n    must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases    k  =  2    and 3, we show that every  n ‐vertex graph with odd girth    2  k  +  1    and minimum degree bigger than3   4  k    n    is homomorphic to the cycle of length    2  k  +  1   . This is best possible in the sense that there are graphs with minimum degree3   4  k    n    and odd girth    2  k  +  1    that are not homomorphic to the cycle of length    2  k  +  1   . Similar results were obtained by Brandt and Ribe‐Baumann.

On the Structure of Graphs with Given Odd Girth and Large Minimum Degree