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

TL;DR

This paper investigates the structure of graphs with a specified odd girth and large minimum degree, establishing conditions under which such graphs are homomorphic to a cycle, extending classical results.

Contribution

It generalizes previous results by showing that graphs with certain minimum degree conditions are homomorphic to cycles of specified length, broadening understanding of graph structure.

Findings

Graphs with odd girth 2k+1 and minimum degree > 3/4k n are homomorphic to the cycle of length 2k+1.

The minimum degree condition of 3/4k n is shown to be optimal.

Results extend classical theorems by Andre1sfai, Erd015s, and Sf3s.

Abstract

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\frac{2}{2k+1}n$ must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of H\"aggkvist and of H\"aggkvist 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 $\frac{3}{4k}n$ is homomorphic to the cycle of length $2k+1$. This is best possible in the sense that there are graphs with minimum degree $\frac{3}{4k}n$ 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.