The mod $k$ chromatic index of graphs is $O(k)$

Let $chi_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 pmod k$. Scott [{em Discrete Math. 175}, 1-3 (1997), 289--291] proved that $chi_k(G)leq5k^2log k$, and thus settled a question of Pyber [{em Sets, graphs and numbers} (1992), pp. 583--610], who had asked whether $chi_k(G)$ can be bounded solely as a function of $k$. We prove that $chi_k(G)=O(k)$, answering affirmatively a question of Scott.