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

2026-03-16

AI Summary

This work investigates the strong chromatic index of $K_{t,t}$-free graphs with maximum degree $d$, verifying and substantially improving Mahdian’s conjecture on its asymptotic upper bound. By integrating the Kővári–Sós–Turán theorem into the Rödl nibble method for the first time, and combining it with Talagrand’s inequality and a novel concentration technique tailored for small vertex sets, the authors uncover the sparse structure of the square of the line graph, $L(G)^2$. This leads to the bound $\chi'_s(G) \leq (1+o(1))\frac{d^2}{\log d}$, markedly better than the previous best-known $(2+o(1))\frac{d^2}{\log d}$ and likely optimal within the current framework. The approach also yields a general list-coloring result of independent interest.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A strong edge coloring of a graph $G$ is an edge coloring $φ\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $χ'_s(G)$, is the minimum number of colors needed for a strong edge coloring of $G$. Erdős and Nešetřil conjectured in 1985 that if $G$ has maximum degree $d$, then $χ'_s(G) \leq \frac 54 d^2$. Mahdian showed in 2000 that if $G$ is $C_4$-free, then $χ'_s(G) \leq (2+o(1)) \frac{d^2}{\log d}$, 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 $χ'_s(G) \leq (1+o(1)) \frac{d^2}{\log d}$. We employ a variant of the Rödl nibble method to prove this result. The key new ingredient in our adaptation of the method is an application of the Kővári-Sós-Turán 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.

Problem

• strong chromatic index
• K_{t,t}-free graphs
• strong edge coloring
• maximum degree
• induced matching

Innovation

• strong edge coloring
• K_{t,t}-free graphs
• Rödl nibble
• Kővári–Sós–Turán theorem
• Talagrand's inequality