On Modular Edge Colourings of Graphs
Abstract
Given a graph G and an integer k≥ 2, let '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 '2(G) ≤ 4 for every graph G, and posed the question of whether 'k(G) can be bounded solely in terms of k for every k≥ 3. This question was answered in 1997 by Scott, who showed that 'k(G)≤5k2 k, and further asked whether 'k(G) = O(k). Recently, Botler, Colucci, and Kohayakawa (2023) answered Scott's question affirmatively proving that 'k(G) ≤ 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 'k(G) ≤ 177k - 93. In this paper, we further improve the multiplicative constant to 9. More specifically, we prove that there is a function f∈ o(k) for which 'k(G) ≤ 7k + f(k) if k is odd, and 'k(G) ≤ 9k + f(k) if k is even. In doing so, we prove that 'k(G) ≤ k + O(d) for every d-degenerate graph G, which plays a central role in our proof.
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