On the mod $k$ chromatic index of graphs

For a graph $G$ and an integer $k\ge 2$, a ${\chi }_k^{\mathrm{\prime }}$-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 $1(\mathrm{m}\mathrm{o}\mathrm{d}k)$, and ${\chi }_k^{\mathrm{\prime }}(G)$ is the minimum number of colors in a ${\chi }_k^{\mathrm{\prime }}$-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 ${\chi }_k^{\mathrm{\prime }}(G)\le 198k-101$ for every graph $G$. In this paper, we show that ${\chi }_k^{\mathrm{\prime }}(G)\le 177k-93$.