On Modular Edge Colourings of Graphs

2025-07-06

AI Summary

This paper studies the mod k edge chromatic number χ′ₖ(G) of a graph G—the minimum number of colors required to edge-color G such that, in each color class, every non-isolated vertex has degree congruent to 1 modulo k. Prior to this work, the best known upper bound was χ′ₖ(G) ≤ 177k − 93 (Nweit & Yang, 2024). We significantly improve this bound, establishing χ′ₖ(G) ≤ 7k + f(k) for odd k and ≤ 9k + f(k) for even k, where f ∈ o(k); we further extend the result to d-degenerate graphs, yielding χ′ₖ(G) ≤ k + O(d). Our approach combines probabilistic methods, degeneracy-based graph decomposition, inductive construction, and fine-grained control over degree distributions within color classes. This is the first bound with multiplicative constant 9—marking a substantial step toward Scott’s conjecture that χ′ₖ(G) = O(k)—and greatly enhances the asymptotic tightness of upper bounds for mod k edge coloring.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given a graph $G$ and an integer $kgeq 2$, let $χ'_k(G)$ denote the minimum number of colours required to colour the edges of $G$ such that, in each colour class, the subgraph induced by the edges of that colour has all non-zero degrees congruent to $1$ modulo $k$. In 1992, Pyber proved that $χ'_2(G) leq 4$ for every graph $G$, and posed the question of whether $χ'_k(G)$ can be bounded solely in terms of $k$ for every $kgeq 3$. This question was answered in 1997 by Scott, who showed that $χ'_k(G)leq5k^2log k$, and further asked whether $χ'_k(G) = O(k)$. Recently, Botler, Colucci, and Kohayakawa (2023) answered Scott's question affirmatively proving that $χ'_k(G) leq 198k - 101$, and conjectured that the multiplicative constant could be reduced to $1$. A step towards this latter conjecture was made in 2024 by Nweit and Yang, who improved the bound to $χ'_k(G) leq 177k - 93$. In this paper, we further improve the multiplicative constant to $9$. More specifically, we prove that there is a function $fin o(k)$ for which $χ'_k(G) leq 7k + f(k)$ if $k$ is odd, and $χ'_k(G) leq 9k + f(k)$ if $k$ is even. In doing so, we prove that $χ'_k(G) leq k + O(d)$ for every $d$-degenerate graph $G$, which plays a central role in our proof.

Problem

• Determine upper bounds for modular edge coloring in graphs.
• Improve multiplicative constants in existing coloring bounds.
• Address conjectures on optimal modular edge coloring.

Innovation

• Improved edge coloring with modulo constraints
• Reduced multiplicative constant to 9
• Used degenerate graphs for proof optimization