  A graph $G$ is said to satisfy the Vizing bound if $\chi(G)\leq \omega(G)+1$,
where $\chi(G)$ and $\omega(G)$ 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.
