On the structure of graphs with given odd girth and large minimum degree

Abstract

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 n-vertex 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.