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

A graph $G$ is said to satisfy the Vizing bound if $χ(G)\leq ω(G)+1$, where $χ(G)$ and $ω(G)$ denote the chromatic number and clique number of $G$, respectively. It was conjectured by Randerath in 1998 that if $G$ is a triangle-free and fork-free graph, where the fork (also known as trident) is obtained from $K_{1,4}$ by subdividing two edges, then $G$ satisfies the Vizing bound. In this paper, we confirm this conjecture.