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

TL;DR

This paper proves that triangle-free and fork-free graphs satisfy the Vizing bound, confirming a conjecture that such graphs have chromatic number at most one more than their clique number.

Contribution

The paper confirms Randerath's 1998 conjecture that triangle-free and fork-free graphs satisfy the Vizing bound, advancing understanding of graph coloring constraints.

Findings

Confirmed the conjecture for triangle-free, fork-free graphs

Established that these graphs satisfy $ ext{chromatic number} \\leq ext{clique number} + 1$

Contributed to graph coloring theory by validating a long-standing hypothesis

Abstract

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.