A stronger bound for the strong chromatic index

TL;DR

This paper establishes a tighter upper bound on the strong chromatic index for large-degree graphs, improving previous results, and introduces a Talagrand-type inequality with a novel exclusion method.

Contribution

It provides a new bound for the strong chromatic index and a modified Talagrand inequality allowing exclusion of unlikely outcomes.

Findings

Proves $oxed{ ext{χ}_s'(G) ext{ } extless= 1.93 ext{ } ext{Δ}(G)^2}$ for large Δ(G).

Improves upon the classical bound by Molloy and Reed.

Introduces a Talagrand-type inequality with outcome exclusions.

Abstract

We prove $\chi_s'(G)\leq 1.93 \Delta(G)^2$ for graphs of sufficiently large maximum degree where $\chi_s'(G)$ is the strong chromatic index of $G$. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where it is allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.