On Modular Edge Colourings of Graphs
Ga\'etan Berthe, Marthe Bonamy, F\'abio Botler, Gaia Carenini, Lucas Colucci, Arthur Dumas, Fatemeh Ghasemi, Pedro Mariano Viana Neto

TL;DR
This paper improves bounds on the minimum number of colours needed for modular edge colourings of graphs, showing it can be linearly bounded in terms of the modulus with a small additive function, advancing understanding of graph colourings.
Contribution
The paper significantly reduces the multiplicative constant in bounds for modular edge colourings, establishing a near-linear bound with a small additive function, and introduces new bounds for d-degenerate graphs.
Findings
Bound $oldsymbol{oldsymbol{ ext{chi}}'_k(G)}$ by $oldsymbol{k + O(d)}$ for $d$-degenerate graphs.
Established bounds $oldsymbol{oldsymbol{ ext{chi}}'_k(G)} o 7k$ or $9k$ depending on parity of $k$.
Improved previous bounds from $198k - 101$ to $9k + f(k)$ with $f o 0$ as $k o ext{large}$.
Abstract
Given a graph and an integer , let denote the minimum number of colours required to colour the edges of such that, in each colour class, the subgraph induced by the edges of that colour has all non-zero degrees congruent to modulo . In 1992, Pyber proved that for every graph , and posed the question of whether can be bounded solely in terms of for every . This question was answered in 1997 by Scott, who showed that , and further asked whether . Recently, Botler, Colucci, and Kohayakawa (2023) answered Scott's question affirmatively proving that , and conjectured that the multiplicative constant could be reduced to . A step towards this latter conjecture was made in 2024 by Nweit and Yang, who improved the bound to $\chi'_k(G)…
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Taxonomy
TopicsGraph Labeling and Dimension Problems
