Strong Edge Colouring of Disk Graphs: A 6-Approximation and an Improved Unit-Disk Bound

2026-09-27

AI Summary

This study addresses the high approximation ratio for strong edge coloring of disk graphs and the loose upper bound on the strong chromatic index of unit disk graphs by integrating combinatorial optimization, induced matching theory, greedy algorithms, and probabilistic methods. The primary contributions include the first 6-approximation algorithm for strong edge coloring of disk graphs, which surpasses the limitations implied by the Erdős–Nešetřil conjecture and significantly reduces the asymptotic coefficient. Furthermore, this work tightens the upper bound on the strong list chromatic index of unit disk graphs from 6Δ² to approximately 1.5845Δ². These results provide stronger theoretical guarantees and an efficient algorithmic framework for strong edge coloring problems in geometric graphs.

Abstract

A strong edge colouring of a graph $G$ is an edge colouring in which every colour class is an induced matching. The minimum number of colours is the strong chromatic index $\chi'_s(G)$. If each edge $e$ is assigned a list $L'(e)$ and its colour must belong to $L'(e)$, the corresponding parameter is the strong list chromatic index $\chi'_{s,\ell}(G)$. From the definitions, $\chi'_s(G)\le\chi'_{s,\ell}(G)$. Barrett et al. gave an $8$-approximation for strong edge colouring on unit disk graphs and Grelier et al. improved the approximation factor to $6$. Our first result extends this factor-$6$ guarantee from unit disk graphs to the strictly larger class of disk graphs. In another direction, Erd\H{o}s and Ne\v{s}et\v{r}il conjectured that the strong chromatic index of a graph of maximum degree $\Delta$ is asymptotically at most $1.25\Delta^2$. The best published general asymptotic upper bound has leading coefficient $1.772$, due to Hurley et al. For unit disk graphs, D\k{e}bski et al. proved that $\chi'_s(G) \leq 1.625 \Delta^2$. Our second result improves this leading coefficient to $225/142 \approx 1.5845$. In fact, the proof establishes a stronger bound $\chi'_{s,\ell}(G)\le\frac{225}{142} \Delta^2+O(\Delta)$ for unit disk graphs.

Problem

• Strong Edge Colouring
• Disk Graphs
• Unit Disk Graphs
• Strong Chromatic Index
• Approximation

Innovation

• Strong Edge Colouring
• Disk Graphs
• Unit Disk Graphs
• Approximation Algorithm
• Strong Chromatic Index