ON THE STRUCTURE OF GRAPHS WITH GIVEN ODD GIRTH AND LARGE MINIMUM DEGREE

Abstract. 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 2k+1 and min-imum degree bigger than 2 2k+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 2k + 1 and minimum degree bigger than 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