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

Let χk′(G) ${\chi }_{k}^{^{\prime} }(G)$ denote the minimum number of colors needed to color the edges of a graph G $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to 1 ( modk) $1\unicode{x02007}(\,\text{mod}\,\,k)$. Scott proved that χk′(G)≤ 5 k 2log k ${\chi }_{k}^{^{\prime} }(G)\le 5{k}^{2}\mathrm{log}\unicode{x0200A}\unicode{x0200A}k$, and thus settled a question of Pyber, who had asked whether χk′(G) ${\chi }_{k}^{^{\prime} }(G)$ can be bounded solely as a function of k $k$. We prove that χk′(G)=O(k) ${\chi }_{k}^{^{\prime} }(G)=O(k)$, answering affirmatively a question of Scott.