List Coloring Triangle-Free Hypergraphs

TL;DR

This paper proves a bound on the list chromatic number of triangle-free hypergraphs of rank three, extending previous results to a more general hypergraph setting with specific degree constraints.

Contribution

It generalizes existing bounds on list coloring from graphs to rank-three hypergraphs with triangle-free conditions and degree limitations.

Findings

List chromatic number bounded by degree ratios and logarithms

Extension of Johansson's and Frieze's results to hypergraphs

Applicable to hypergraphs with maximum degrees nd onstraints

Abstract

A triangle in a hypergraph is a collection of distinct vertices u,v,w and distinct edges e,f,g with u,v \in e, v,w \in f, w,u \in g, and \{u,v,w\} \cap e \cap f \cap g=\emptyset. The i-degree of a vertex in a hypergraph is the number of edges of size i containing it. We prove that every triangle-free hypergraph of rank three (edges have size two or three) with maximum 3-degree \Delta_3 and maximum 2-degree \Delta_2 has list chromatic number at most c max{\Delta_2/ log{\Delta_2}}, (\Delta_3 / log{\Delta_3})^(1/2)} for some absolute positive constant c. This generalizes a result of Johansson and a result of Frieze and the second author.