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

We study the structure of graphs with high minimum degree conditions and given odd girth. For example, the classical work of Andrásfai, Erdős, and Sós implies that every n-vertex graph with odd girth 2k + 1 and minimum degree bigger than 2n 2k+1 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 nvertex graph with odd girth 2k + 1 and minimum degree bigger than 3n 4k is homomorphic to the cycle of length 2k + 1. This is best possible in the sense that there are graphs with minimum degree 3n 4k 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.