Proper conflict-free 6-coloring of planar graphs without short cycles

Abstract

A proper conflict-free l-coloring of a graph G is a proper l-coloring satisfying that for any non-isolated vertex $v\in V\left(G\right)$, there exists a color appearing exactly once in $N_G\left(v\right)$. The proper conflict-free chromatic number, denoted by ${\chi }_{pcf}\left(G\right)$, is the minimal integer l so that G admits a proper conflict-free l-coloring. This notion was proposed by Fabrici et al. in 2022. They focus mainly on proper conflict-free coloring of outerplanar graphs and planar graphs. They constructed a planar graph that has no proper conflict-free 5-coloring and conjectured every planar graph G has ${\chi }_{pcf}\left(G\right)\le 6$. In this paper, we confirm this conjecture for planar graphs without cycles of lengths 3, 5 or 6.