Triangle-free Graphs with Large Minimum Common Degree

Abstract

Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G):xy\in E(G)\}$. The minimum common degree of $G$, denoted by $\hat{I}{´}_2(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, Häggkvist showed that every triangle-free graph with minimum degree greater than $\lfloor \frac{3n}{8}\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\lfloor \frac{n}{8}\rfloor$ is homomorphic to a cycle of length 5, which implies Häggkvist's result. The balanced blow-up of the Möbius ladder graph shows that it is best possible.

11 pages, 9 figures