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

Abstract

Let ${\chi }_k^{\prime }(G)$ 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 $1(\mathrm{mod}k)$. Scott [{\em Discrete Math. 175}, 1-3 (1997), 289--291] proved that ${\chi }_k^{\prime }(G)\le 5k^2\log k$, and thus settled a question of Pyber [{\em Sets, graphs and numbers} (1992), pp. 583--610], who had asked whether ${\chi }_k^{\prime }(G)$ can be bounded solely as a function of $k$. We prove that ${\chi }_k^{\prime }(G)=O(k)$, answering affirmatively a question of Scott.

3 pages