The $\mathrm{mod}k$ chromatic index of random graphs

Abstract

The $\mathrm{mod}k$ 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 $1(\mathrm{mod}k)$. Recently, the authors proved that the $\mathrm{mod}k$ 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 $\mathrm{mod}k$ chromatic index of random graphs. We prove that for every integer $k\ge 2$, there is $C_k>0$ such that if $p\ge C_k{n}^{-1}\log n$ and $n(1-p)\to \infty$ as $n\to \infty$, then the following holds: if $k$ is odd, then the $\mathrm{mod}k$ chromatic index of $G(n,p)$ is asymptotically almost surely equal to $k$, while if $k$ is even, then the $\mathrm{mod}k$ chromatic index of $G(2n,p)$ (respectively $G(2n+1,p)$) is asymptotically almost surely equal to $k$ (respectively $k+1$).

arXiv comment12 pages, 1 figure. To appear in J. of Graph Theory