Single conflict coloring and palette sparsification of uniform hypergraphs

TL;DR

This paper introduces single conflict coloring for r-uniform hypergraphs, analyzes probabilistic models, and establishes thresholds and conditions for colorability, contributing new theoretical insights into hypergraph coloring.

Contribution

It presents the concept of single conflict coloring, studies probabilistic thresholds, and provides palette sparsification results for linear uniform hypergraphs.

Findings

Sharp threshold for complete graphs in the probabilistic model

Sufficient condition for single conflict colorability in hypergraphs

Palette sparsification results for linear uniform hypergraphs

Abstract

We introduce and investigate single conflict coloring in the setting of r-uniform hypergraphs. We establish some basic properties of this hypergraph coloring model and study a probabilistic model of single conflict coloring where the conflicts for each edge are chosen randomly; in particular, we prove a sharp threshold-type result for complete graphs and establish a sufficient condition for single conflict colorability of r-uniform hypergraphs in this model. Furthermore, we obtain a related palette sparsification-type result for general list coloring of linear uniform hypergraphs (i.e. uniform hypergraphs where any two edges share at most one common vertex). Throughout the paper we pose several questions and conjectures