Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Modular Chromatic Index of Random Hypergraphs

Gaia Carenini, Samuel Coulomb

TL;DR

This work introduces and resolves the modular $k$-chromatic index for random $r$-uniform hypergraphs, showing that for $H_{n,p,r}$ with $p o (C\log n)/n$ and $n^{r-1}p(1-p) o\infty$, the a.a.s. value of $ abla_k'(H_{n,p,r})$ equals $k$ when $n\equiv 0\pmod{\gcd(k,r)}$ and lies in the range $[\max(k,r),k+r+1]$ otherwise. A key technical contribution is a robust sufficient condition guaranteeing the existence of a $k$-factor in subhypergraphs of $H_{n,p,r}$, enabling the decomposition into $1_k$-subhypergraphs. The proof adapts and extends methods from the graph setting (e.g., Hall’s theorem, McDiarmid’s inequality) to hypergraphs, including careful random partitions and concentration arguments. The results extend the graph-based theory of mod-$k$ decompositions to hypergraphs and address questions about linear bounds on $ abla_k'$. Overall, the paper provides a concrete probabilistic construction and a structural factor condition that together determine the modular edge-coloring behavior of typical 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 $χ_k'(H)$ 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 $χ_k'(H) = k$ if $n$ is divisible by $\gcd(k,r)$, and $\max(k,r) \le χ_k'(H) \le k+r+1$ 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, Peña, and Ueckerdt (2025) regarding whether $χ_2'(H)$ can be bounded by a constant for every hypergraph $H$.

On the Modular Chromatic Index of Random Hypergraphs

TL;DR

This work introduces and resolves the modular -chromatic index for random -uniform hypergraphs, showing that for with and , the a.a.s. value of equals when and lies in the range otherwise. A key technical contribution is a robust sufficient condition guaranteeing the existence of a -factor in subhypergraphs of , enabling the decomposition into -subhypergraphs. The proof adapts and extends methods from the graph setting (e.g., Hall’s theorem, McDiarmid’s inequality) to hypergraphs, including careful random partitions and concentration arguments. The results extend the graph-based theory of mod- decompositions to hypergraphs and address questions about linear bounds on . Overall, the paper provides a concrete probabilistic construction and a structural factor condition that together determine the modular edge-coloring behavior of typical random hypergraphs.

Abstract

Let be two integers. We consider the problem of partitioning the hyperedge set of an -uniform hypergraph into the minimum number of edge-disjoint subhypergraphs in which every vertex has either degree or degree congruent to modulo . For a random hypergraph drawn from the binomial model , with edge probability for a large enough constant independent of and satisfying as , we show that asymptotically almost surely if is divisible by , and otherwise. A key ingredient in our approach is a sufficient condition ensuring the existence of a -factor, a -regular spanning subhypergraph, within subhypergraphs of a random hypergraph from , 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, Peña, and Ueckerdt (2025) regarding whether can be bounded by a constant for every hypergraph .

Paper Structure

This paper contains 7 sections, 6 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Proof Overview
  4. Factors in Subhypergraphs of Random Hypergraphs
  5. Proof of the Main Theorem (Theorem \ref{['thrm:main']})
  6. Case 1: $n \equiv 0 \pmod d$.
  7. Case 2: $n \not\equiv 0 \pmod d$.

Key Result

Theorem 1

Let $k\ge2$ and $r\ge2$ be fixed integers, and let $d$ denote $\gcd(k,r)$. There exists a positive constant $C$ such that for every integer $n \ge 1$ and all $p \in \left(C\log(n)/n,1\right)$ such that $n^{r-1}p(1-p) \rightarrow \infty$ as $n \rightarrow \infty$, a.a.s. the mod $k$ chromatic index o

Theorems & Definitions (8)

  • Theorem 1: Main Theorem
  • Theorem 2: Main Technical Theorem
  • Lemma 3: Chernoff Bound
  • Theorem 4: McDiarmid Bound
  • Theorem 5: Hall's Theorem
  • proof : Proof of Theorem \ref{['thrm:factor']}
  • Lemma 6
  • proof : Proof of Theorem \ref{['thrm:main']}