On the Modular Chromatic Index of Random Hypergraphs

Abstract

Let $k,r \geq 2$ be two integers. We consider the problem of partitioning the hyperedge set of an $r$-uniform hypergraph $H$ into the minimum number $\chi_k'(H)$ of edge-disjoint subhypergraphs in which every vertex has either degree $0$ or degree congruent to $1$ modulo $k$. For a random hypergraph $H$ drawn from the binomial model $\mathbf{H}(n,p,r)$, with edge probability $p \in (C\log(n)/n,1)$ for a large enough constant $C>0$ independent of $n$ and satisfying $n^{r-1}p(1-p)\to\infty$ as $n\to\infty$, we show that asymptotically almost surely $\chi_k'(H) = k$ if $n$ is divisible by $\gcd(k,r)$, and $\max(k,r) \le \chi_k'(H) \le k+r+1$ otherwise. A key ingredient in our approach is a sufficient condition ensuring the existence of a $k$-factor, a $k$-regular spanning subhypergraph, within subhypergraphs of a random hypergraph from $\mathbf{H}(n,p,r)$, a result that may be of independent interest. Our main result extends a theorem of Botler, Colucci, and Kohayakawa (2023), who proved an analogous statement for graphs, and provides a partial answer to a question posed by Goetze, Klute, Knauer, Parada, Pe\~na, and Ueckerdt (2025) regarding whether $\chi_2'(H)$ can be bounded by a constant for every hypergraph $H$.