Proper conflict-free coloring of sparse graphs

TL;DR

This paper determines the maximum average degree thresholds for proper conflict-free c-colorings in graphs, providing tight bounds and examples, and extends results to planar graphs with girth at least 5.

Contribution

It establishes exact maximum average degree bounds guaranteeing proper conflict-free colorings for all c, including tightness examples and special cases for planar graphs.

Findings

Graphs with mad(G) ≤ 4c/(c+2) have proper conflict-free c-colorings for c ≥ 5.

Graphs with mad(G) < 12/5 have proper conflict-free 4-colorings.

Planar graphs with girth ≥ 5 have proper conflict-free 7-colorings.

Abstract

A {\it proper conflict-free $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood. This notion was formally introduced by Fabrici et al., who proved that planar graphs have a proper conflict-free 8-coloring and constructed a planar graph with no proper conflict-free 5-coloring. Caro, Petru\v{s}evski, and \v{S}krekovski investigated this coloring concept further, and in particular studied upper bounds on the maximum average degree that guarantees a proper conflict-free $c$-coloring for $c\in\{4,5,6\}$. Along these lines, we completely determine the threshold on the maximum average degree of a graph $G$, denoted $mad(G)$, that guarantees a proper conflict-free $c$-coloring for all $c$ and also provide tightness examples. Namely, for $c\geq 5$ we prove that a graph $G$ with $mad(G)\leq \frac{4c}{c+2}$ has a proper conflict-free $c$-coloring, unless $G$ contains a $1$-subdivision of the complete graph on $c+1$ vertices. When $c=4$, we show that a graph $G$ with $mad(G)<\frac{12}{5}$ has a proper conflict-free $4$-coloring, unless $G$ contains an induced $5$-cycle. In addition, we show that a planar graph with girth at least 5 has a proper conflict-free $7$-coloring.