For a graph $G$ and an integer $k\geq 2$, a $\chi'_{k}$-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 $1 ~ (\mod k)$, and $\chi'_{k}(G)$ is the minimum number of colors in a $\chi'_{k}$-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 $\chi'_{k}(G)\leq 198k-101$ for every graph $G$. In this paper, we show that $\chi'_{k}(G) \leq 177k-93$.
Discrete Mathematics & Theoretical Computer Science