On the 3-colorability of triangle-free and fork-free graphs

Joshua Schroeder, Zhiyu Wang, Xingxing Yu

A graph $G$ is said to satisfy the Vizing bound if $\ddot{I}‡(G)\le \ddot{I}‰(G)+1$, where $\ddot{I}‡(G)$ and $\ddot{I}‰(G)$ denote the chromatic number and clique number of $G$, respectively. It was conject…