On the mod k chromatic index of graphs

For a graph G and an integer k≥ 2, a χ'ₖ-coloring of G is an edge coloring of G such that the subgraph induced by the edges of each color has all degrees congruent to 1 ~ ( k), and χ'ₖ(G) is the minimum number of colors in a χ'ₖ-coloring of G. In ["The mod k chromatic index of graphs is $O(k)$", J. Graph Theory. 2023; 102: 197-200], Botler, Colucci and Kohayakawa proved that χ'ₖ(G)≤ 198k-101 for every graph G. In this paper, we show that χ'ₖ(G) ≤ 177k-93.