Proper conflict-free choosability of planar graphs

TL;DR

This paper proves new bounds on proper conflict-free list coloring for specific classes of planar graphs, confirming two conjectures and establishing that certain graphs are proper conflict-free 6-choosable.

Contribution

It confirms two conjectures on proper conflict-free choosability for planar and related graphs, providing bounds and extending known results.

Findings

Confirmed the second conjecture for specific graph classes.

Confirmed the first conjecture for these classes and all outer-1-planar graphs.

Established that certain planar and outer-1-planar graphs are proper conflict-free 6-choosable.

Abstract

A proper conflict-free coloring of a graph is a proper vertex coloring wherein each non-isolated vertex's open neighborhood contains at least one color appearing exactly once. For a non-negative integer $k$, a graph $G$ is said to be proper conflict-free (degree+$k$)-choosable if given any list assignment $L$ for $G$ where $|L(v)| = d(v) + k$ holds for every vertex $v \in V(G)$, there exists a proper conflict-free coloring $\phi$ of $G$ such that $\phi(v) \in L(v)$ for all $v \in V(G)$. Recently, Kashima, \v{S}krekovski, and Xu proposed two related conjectures on proper conflict-free choosability: the first asserts the existence of an absolute constant $k$ such that every graph is proper conflict-free (degree+$k$)-choosable, while the second strengthens this claim by restricting to connected graphs other than the cycle of length 5 and reducing the constant to $k=2$. In this paper, we confirm the second conjecture for three graph classes: $K_4$-minor-free graphs with maximum degree at most 4, outer-1-planar graphs with maximum degree at most 4, and planar graphs with girth at least 12; we also confirm the first conjecture for these same graph classes, in addition to all outer-1-planar graphs (without degree constraints). Moreover, we prove that planar graphs with girth at least 12 and outer-1-planar graphs are proper conflict-free $6$-choosable.