Proper conflict-free choosability of planar graphs

TL;DR

This work advances the theory of proper conflict-free (PCF) coloring by proving PCF-$(\text{degree}+2)$-choosability for three restricted yet broad classes: connected $K_4$-minor-free graphs with $\Delta\le 4$, outer-$1$-planar graphs with $\Delta\le 4$, and planar graphs with girth at least $12$. It introduces a comprehensive reducibility framework based on a large catalog of unavailable configurations and extension obstacles, then shows that each critical configuration $T_i$ is reducible, ruling out minimal counterexamples. Consequently, the paper confirms a strengthened form of the conjectures for these classes (and related cases), and also establishes PCF-$(\text{degree}+2)$-choosability in the same families, with PCF-6-choosability for planar girth-12 and outer-$1$-planar graphs. The results deepen our understanding of conflict-free colorings in planar-related graph classes and provide a roadmap for extending PCF-choosability to broader settings. They also sharpen known bounds and connect to broader questions about $(2\cdot\text{degree}+1)$-choosability and asymptotic behavior in dense graphs.

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 $φ$ of $G$ such that $φ(v) \in L(v)$ for all $v \in V(G)$. Recently, Kashima, Škrekovski, 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.

Figures (3)

• Figure 1: Unavailable configurations
• Figure 2: Three frequently occurring configurations
• Figure 3: Three frequently occurring configurations

Theorems & Definitions (55)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1