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Balanced colorings of Erdős-Rényi hypergraphs

Abhishek Dhawan, Yuzhou Wang

TL;DR

This work determines the asymptotic balanced chromatic number $\chi_b(H)$ for sparse $r$-uniform, $r$-partite Erdős--Rényi hypergraphs $H \sim \mathcal{H}(r,n,p)$ with $p = d/n^{r-1}$. The authors extend Feige–Kogan-type bounds to multipartite hypergraphs by proving that any balanced colorable hypergraph of average degree $d$ admits a balanced coloring with at most $r(r-1)d+1$ colors, and then implement a three-step probabilistic coloring scheme (almost coloring, expose-and-merge, color small subsets) to color almost all vertices with $q = \left((r-1)/r \cdot d/\log d\right)^{1/(r-1)}$ colors, completing the remainder with a few more colors. They establish that, with high probability, $\chi_b(H)$ lies in the interval $\big( (1-\varepsilon)ig/\big( (r-1)/r \cdot d/\log d \big)^{1/(r-1)}, (1+\varepsilon)ig( (r-1)/r \cdot d/\log d \big)^{1/(r-1)} \big)$ for all fixed $\varepsilon>0$, as $d$ grows, generalizing known bipartite results to all $r\ge 2$. The methods combine a second-moment analysis, a Matula-style exposure process, and reductions to perfect-matchings in the multipartite complement, yielding tight thresholds and shedding light on the distinct behavior of balanced colorings compared to ordinary colorings in hypergraphs.

Abstract

An $r$-uniform hypergraph $H = (V, E)$ is $r$-partite if there exists a partition of the vertex set into $r$ parts such that each edge contains exactly one vertex from each part. We say an independent set in such a hypergraph is balanced if it contains an equal number of vertices from each partition. The balanced chromatic number of $H$ is the minimum value $q$ such that $H$ admits a proper $q$-coloring where each color class is a balanced independent set. In this note, we determine the asymptotic behavior of the balanced chromatic number for sparse $r$-uniform $r$-partite Erdős--Rényi hypergraphs. A key step in our proof is to show that any balanced colorable hypergraph of average degree $d$ admits a proper balanced coloring with $r(r-1)d + 1$ colors. This extends a result of Feige and Kogan on bipartite graphs to this more general setting.

Balanced colorings of Erdős-Rényi hypergraphs

TL;DR

This work determines the asymptotic balanced chromatic number for sparse -uniform, -partite Erdős--Rényi hypergraphs with . The authors extend Feige–Kogan-type bounds to multipartite hypergraphs by proving that any balanced colorable hypergraph of average degree admits a balanced coloring with at most colors, and then implement a three-step probabilistic coloring scheme (almost coloring, expose-and-merge, color small subsets) to color almost all vertices with colors, completing the remainder with a few more colors. They establish that, with high probability, lies in the interval for all fixed , as grows, generalizing known bipartite results to all . The methods combine a second-moment analysis, a Matula-style exposure process, and reductions to perfect-matchings in the multipartite complement, yielding tight thresholds and shedding light on the distinct behavior of balanced colorings compared to ordinary colorings in hypergraphs.

Abstract

An -uniform hypergraph is -partite if there exists a partition of the vertex set into parts such that each edge contains exactly one vertex from each part. We say an independent set in such a hypergraph is balanced if it contains an equal number of vertices from each partition. The balanced chromatic number of is the minimum value such that admits a proper -coloring where each color class is a balanced independent set. In this note, we determine the asymptotic behavior of the balanced chromatic number for sparse -uniform -partite Erdős--Rényi hypergraphs. A key step in our proof is to show that any balanced colorable hypergraph of average degree admits a proper balanced coloring with colors. This extends a result of Feige and Kogan on bipartite graphs to this more general setting.

Paper Structure

This paper contains 13 sections, 16 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Background and main result
  3. Proof Overview
  4. Future directions
  5. $\Vec{\bm{\gamma}}$-Balanced Colorings.
  6. Balanced $\mathbf{\beta}$-Colorings.
  7. Two-Point Concentration.
  8. Preliminaries
  9. Proof of Theorem \ref{['theorem: hypergraph version of FK result']}
  10. Proof of Theorem \ref{['theorem: main result']}
  11. Proof of Lemma \ref{['lemma: almost coloring']}
  12. Proof of Lemma \ref{['lemma: expose and merge']}
  13. Proof of Lemma \ref{['lemma: color small subsets']}

Key Result

Theorem 1.3

For all $r \geq 2$ and $\varepsilon \in (0, 1)$, there exists $d_0 \in \mathbb{N}$ such that the following holds for all $d \geq d_0$. There exists $n_0 \in \mathbb{N}$ such that for any $n \geq n_0$ and $H \sim \mathcal{H}(r,n,p)$ for $p = d/n^{r-1}$, we have with high probability.

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: dhawan2023balanced
  • Lemma 2.3
  • proof
  • Theorem 2.4: Talagrand's Inequality; molloy2014colouring
  • Theorem : Restatement of Theorem \ref{['theorem: hypergraph version of FK result']}
  • ...and 29 more