Triangle-free Graphs with Large Minimum Common Degree

Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\text{\hspace{0.17em}⁣}:xy\in E(G)\}$. The minimum common degree of $G$, denoted by ${\delta }_2(G)$, is defined as the minimum of $\mathrm{\mid }N(x)\cap N(y)\mathrm{\mid }$ 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.