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

TL;DR

This paper proves that the mod $k$ chromatic index of graphs can be bounded linearly in $k$, improving previous bounds and answering a longstanding open question.

Contribution

It establishes that the mod $k$ chromatic index is $O(k)$, providing a linear bound and resolving a question posed by Scott.

Findings

The mod $k$ chromatic index is bounded above by a linear function of $k$.

Previous bounds were polynomial in $k$, specifically $O(k^2 ext{log} k)$.

The result confirms that the index depends only on $k$, not on the graph size.

Abstract

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 $\chi'_k(G)\leq5k^2\log k$, and thus settled a question of Pyber [{\em Sets, graphs and numbers} (1992), pp. 583--610], who had asked whether $\chi_k'(G)$ can be bounded solely as a function of $k$. We prove that $\chi'_k(G)=O(k)$, answering affirmatively a question of Scott.