The strong chromatic index of $K_{t,t}$-free graphs

TL;DR

This paper proves a conjecture that the strong chromatic index of $K_{t,t}$-free graphs with maximum degree $d$ is at most approximately $rac{d^2}{ ext{log } d}$, using advanced probabilistic and combinatorial methods.

Contribution

The authors confirm Mahdian's conjecture and improve the upper bound for the strong chromatic index of $K_{t,t}$-free graphs, introducing novel structural analysis via the Rödl nibble method.

Findings

Established the bound $oxed{ ext{strong chromatic index} \, ilde{<} \, rac{d^2}{ ext{log } d}}$ for $K_{t,t}$-free graphs.

Applied the Kővári-Sós-Turán theorem to analyze the structure of the square of the line graph.

Developed a new structural property for list coloring that may be of independent interest.

Abstract

A strong edge coloring of a graph $G$ is an edge coloring $\phi\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $\chi'_s(G)$, is the minimum number of colors needed for a strong edge coloring of $G$. Erd\H{o}s and Ne\v{s}et\v{r}il conjectured in 1985 that if $G$ has maximum degree $d$, then $\chi'_s(G) \leq \frac 54 d^2$. Mahdian showed in 2000 that if $G$ is $C_4$-free, then $\chi'_s(G) \leq (2+o(1)) \frac{d^2}{\log d}$, and he conjectured that the same upper bound holds for $K_{t,t}$-free graphs. In this paper, we prove this conjecture and improve upon it to show the following: every $K_{t,t}$-free graph $G$ of maximum degree $d$ satisfies $\chi'_s(G) \leq (1+o(1)) \frac{d^2}{\log d}$. We employ a variant of the R\"odl nibble method to prove this result. The key new ingredient in our adaptation of the method is an application of the K\H{o}v\'ari-S\'os-Tur\'an theorem to show that $H := L(G)^2$ satisfies certain structural properties. These properties, in conjunction with a variant of Talagrand's inequality to handle exceptional outcomes, allow us to concentrate the sizes of certain vertex sets through the nibble, even when these vertex sets have order smaller than the maximum codegree of $H$. We encapsulate these structural properties into a more general statement on list coloring that we believe to be of independent interest. In light of the conjectured computational threshold for coloring random graphs arising in average-case complexity theory, we suspect that our result is best possible using this approach.