Single conflict coloring and palette sparsification of 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