On Vizing's problem for triangle-free graphs

TL;DR

This paper advances Vizing's problem by proving an improved chromatic number bound for large maximum degree triangle-free graphs using novel counting methods.

Contribution

It introduces a new counting argument and applies Hurley and Pirot's method to establish a tighter bound for triangle-free graphs with high maximum degree.

Findings

Proves $ ext{chi}(G) extless= igl\lceil ( ext{Delta}+1)/2 igr ceil + 1$ for $ ext{Delta} extgreater= 524$

Provides progress towards Vizing's conjecture for large $ ext{Delta}$

Utilizes a novel counting argument in graph coloring theory

Abstract

We prove that $\chi(G) \le \lceil (\Delta+1)/2\rceil+1$ for any triangle-free graph $G$ of maximum degree $\Delta$ provided $\Delta \ge 524$. This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a method of Hurley and Pirot, which in turn relies on a new counting argument of the second author.