Proper Conflict-free Coloring of Graphs with Large Maximum Degree

Abstract

A proper coloring of a graph is conflict-free if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petruševski, and Škrekovski proved that every graph G has a proper conflict-free coloring with at most 5∆(G)/2 colors and conjectured that ∆(G) + 1 colors suffice for every connected graph G with ∆(G) 3. Our first main result is that even for list-coloring, 1.6550826∆(G) + ∆(G) colors suffice for every graph G with ∆(G) 10 9 ; we also prove slightly weaker bounds for all graphs with ∆(G) 750. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph G and a "conflict" hypergraph H. As another corollary of our results in this general framework, every graph has a proper ( √ 30 + o(1))∆(G) 1.5 -list-coloring such that every bi-chromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lovász Local Lemma or entropy compression.We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph G has a fractional (1 + o(1))∆(G)-coloring such that every fractionally bi-chromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro et al. holds asymptotically in a strong sense.