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List colorings of $k$-partite $k$-graphs

Abhishek Dhawan

TL;DR

This work resolves list-coloring questions for $k$-partite $k$-graphs by establishing a near-optimal threshold: for large maximum degree $\Delta$, every vertex list of size at least $\left((k-1+\varepsilon)\Delta/\log \Delta\right)^{1/(k-1)}$ yields a proper $L$-coloring. The authors develop a general framework based on per-vertex degree, color-degree, and list sizes, leveraging the Lovász Local Lemma to extend partial colorings to whole hypergraphs, and they extend the results to DP-coloring (correspondence coloring) with asymmetric variants. They also connect to known bounds for the chromatic numbers of simple and triangle-free hypergraphs, matching these up to constant factors, and provide corollaries covering color-degree and asymmetric scenarios. The methods offer a versatile approach for hypergraph coloring under list constraints and open avenues for extensions to related coloring models.

Abstract

A $k$-uniform hypergraph (or $k$-graph) $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. In this note, we consider list colorings for such hypergraphs. We show that for any $\varepsilon > 0$ if each vertex $v \in V(H)$ is assigned a list of size $|L(v)| \geq \left((k-1+\varepsilon)Δ/\log Δ\right)^{1/(k-1)}$, then $H$ admits a proper $L$-coloring, provided $Δ$ is sufficiently large. Up to a constant factor, this matches the bound on the chromatic number of simple $k$-graphs shown by Frieze and Mubayi, and that on the list chromatic number of triangle free $k$-graphs shown by Li and Postle. Our results hold in the more general setting of ``color-degree'' as has been considered for graphs. Furthermore, we establish a number of asymmetric statements matching results of Alon, Cambie, and Kang for bipartite graphs.

List colorings of $k$-partite $k$-graphs

TL;DR

This work resolves list-coloring questions for -partite -graphs by establishing a near-optimal threshold: for large maximum degree , every vertex list of size at least yields a proper -coloring. The authors develop a general framework based on per-vertex degree, color-degree, and list sizes, leveraging the Lovász Local Lemma to extend partial colorings to whole hypergraphs, and they extend the results to DP-coloring (correspondence coloring) with asymmetric variants. They also connect to known bounds for the chromatic numbers of simple and triangle-free hypergraphs, matching these up to constant factors, and provide corollaries covering color-degree and asymmetric scenarios. The methods offer a versatile approach for hypergraph coloring under list constraints and open avenues for extensions to related coloring models.

Abstract

A -uniform hypergraph (or -graph) is -partite if can be partitioned into sets such that each edge in contains precisely one vertex from each . In this note, we consider list colorings for such hypergraphs. We show that for any if each vertex is assigned a list of size , then admits a proper -coloring, provided is sufficiently large. Up to a constant factor, this matches the bound on the chromatic number of simple -graphs shown by Frieze and Mubayi, and that on the list chromatic number of triangle free -graphs shown by Li and Postle. Our results hold in the more general setting of ``color-degree'' as has been considered for graphs. Furthermore, we establish a number of asymmetric statements matching results of Alon, Cambie, and Kang for bipartite graphs.
Paper Structure (7 sections, 12 theorems, 43 equations)

This paper contains 7 sections, 12 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Corollaries
  4. Proof of Theorem \ref{['theo:conditions_theo']}
  5. Proof assuming \ref{['cond:1']}
  6. Proof assuming \ref{['cond:3']}
  7. Proof assuming \ref{['cond:2']}

Key Result

Theorem 1.1

For $k \geqslant 3$, the following holds for sufficiently large $\Delta \in {\mathbb{N}}$. Let $H$ be a simple $k$-graph of maximum degree at most $\Delta$. Then for some constant $c \coloneqq c(k) > 0$, we have

Theorems & Definitions (24)

  • Theorem 1.1: frieze2013coloring
  • Theorem 1.2: li2022chromatic
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Conjecture 1.9: alon1998choice
  • Theorem 1.10: meroueh2019list
  • ...and 14 more