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

A graph G is said to satisfy the Vizing bound if χ(G) ≤ 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 K1,4 by subdividing two edges, then G satisfies the Vizing bound. In this paper, we confirm this conjecture.

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