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

TL;DR

The paper resolves Randerath's conjecture by proving that triangle-free and fork-free graphs satisfy the Vizing bound χ(G) ≤ ω(G)+1, i.e., the trident conjecture. The authors employ a minimal counterexample framework centered on a good 5-cycle C, establish a 3-connectedness baseline, and prove a key independent-path constraint for N^2(C). They develop detailed structural properties around good 5-cycles and perform an extensive, case-based analysis—aided by computer-assisted search via Structure, Coloring, and Mixed algorithms—to rule out all potential counterexamples. The result completes the characterization of saturated good Vizing pairs involving K_3 and yields corollaries for paw-free graphs and related forbidden-subgraph families, advancing the understanding of χ-boundedness in triangle- and fork-free graph classes.

Abstract

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.

Figures (1)

• Figure 1: Forbidden graphs

Theorems & Definitions (55)

• Conjecture 1: Randerath RanderathRanderath_the_second
• Theorem 1.1: Fan, Xu, Ye, and Yu yeyu
• Theorem 1.2
• Corollary 1.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof