Résumé
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.