Balanced independent sets and colorings of hypergraphs

TL;DR

We study balanced independent sets and balanced colorings in $n$-balanced $k$-uniform $k$-partite hypergraphs $(V_1\cup\dots\cup V_k, E)$, proving tight asymptotic bounds: an upper bound on the balanced independence number $α_b(H)$ in terms of the average degree $D=|E|/n$, and an upper bound on the balanced chromatic number $χ_b(H)$ in terms of the maximum degree $Δ$. The main techniques are probabilistic: an Erdős–Rényi style hypergraph model $\mathcal{H}(k,N,p)$ yields the upper bound for $α_b(H)$, while a randomized two-stage coloring together with a residual low-degree subhypergraph yields the bound on $χ_b(H)$. The results extend prior graph results (k=2) to general $k$, and align with bounds known for list-coloring and triangle-free hypergraphs, using tools such as Harris's inequality, Chernoff bounds, and Talagrand's inequality. The work opens avenues for studying computational hardness and extensions to list colorings and other structural properties in hypergraphs.

Abstract

A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I\cap V_j|$ for each $1 \leq i, j \leq k$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $α_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $χ_b(H)$ in terms of the maximum degree $Δ$. Our results recover those of recent work of Chakraborti for $k = 2$.

Theorems & Definitions (35)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: axenovich2021bipartite
• Theorem 1.4: chakraborti2023extremal
• Theorem 1.5
• Theorem 1.6: chakraborti2023extremal
• Theorem 1.7
• proof : Proof
• Lemma 2.2
• proof