The $\!{}\bmod k$ chromatic index of random graphs

TL;DR

This paper investigates the $mod k$ chromatic index of random graphs, establishing asymptotic values depending on the parity of $k$ and the size of the graph, extending previous bounds to probabilistic settings.

Contribution

It determines the asymptotic behavior of the $mod k$ chromatic index for random graphs under certain edge probability conditions, generalizing prior deterministic bounds.

Findings

For odd $k$, the index is asymptotically $k$.

For even $k$, the index is asymptotically $k$ or $k+1$ depending on the graph size.

Results hold with high probability under specified edge probability conditions.

Abstract

The $\!{}\bmod k$ chromatic index of a graph $G$ is the minimum number of colors needed to color the edges of $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1\!\!\pmod k$. Recently, the authors proved that the $\!{}\bmod k$ chromatic index of every graph is at most $198k-101$, improving, for large $k$, a result of Scott [Discrete Math. 175, 1-3 (1997), 289-291]. Here we study the $\!{}\bmod k$ chromatic index of random graphs. We prove that for every integer $k\geq2$, there is $C_k>0$ such that if $p\geq C_kn^{-1}\log{n}$ and $n(1-p) \rightarrow\infty$ as $n\to\infty$, then the following holds: if $k$ is odd, then the $\!{}\bmod k$ chromatic index of $G(n,p)$ is asymptotically almost surely equal to $k$, while if $k$ is even, then the $\!{}\bmod k$ chromatic index of $G(2n,p)$ (respectively $G(2n+1,p)$) is asymptotically almost surely equal to $k$ (respectively $k+1$).