On Modular Edge Colourings of Graphs
Gaétan Berthe, Marthe Bonamy, Fábio Botler, Gaia Carenini, Lucas Colucci, Arthur Dumas, Fatemeh Ghasemi, Pedro Mariano Viana Neto
TL;DR
This work advances the study of modular edge colourings by proving near-linear upper bounds on $χ'_k(G)$ that depend only on $k$. The authors reduce the problem to coloring a bounded-degeneracy remainder after removing a maximal $1_k$-subgraph, then apply a strengthened bound on $k$-divisible subgraphs together with a refined Hall-type star-covering argument to colour the remainder. Their main results show $χ'_k(G) ≤ 9k + f(k)$ for even $k$ and $χ'_k(G) ≤ 7k + f(k)$ for odd $k$, with $f(k) ∈ o(k)$, and they establish a general bound $χ'_k(G) ≤ k + O(d)$ for $d$-degenerate graphs. These contributions bring the problem closer to a linear bound in $k$ and highlight the roles of degeneracy and divisibility constraints in modular edge colourings.
Abstract
Given a graph $G$ and an integer $k\geq 2$, 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 $k\geq 3$. This question was answered in 1997 by Scott, who showed that $χ'_k(G)\leq5k^2\log k$, 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 $f\in o(k)$ 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.