A proper conflict-free l-coloring of a graph G is a proper l-coloring satisfying that for any non-isolated vertex v∈V(G), there exists a color appearing exactly once in NG(v). The proper conflict-free chromatic number, denoted by χpcf(G), 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 χpcf(G)≤6. In this paper, we confirm this conjecture for planar graphs without cycles of lengths 3, 5 or 6.

中文翻译：

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平面图的正确无冲突 6 色，无需短周期


图形 G 的适当无冲突 l 着色是适当的 l 着色，满足任何非孤立顶点 v∈V（G） 存在一种颜色在 NG（v） 中恰好出现一次。正确的无冲突色度数（用 χpcf（G） 表示）是最小整数 l，因此 G 允许适当的无冲突 l 着色。这个概念是由 Fabrici 等人在 2022 年提出的。他们主要关注外平面图和平面图的正确无冲突着色。他们构建了一个没有适当的无冲突 5 色的平面图，并推测每个平面图 G 都有 χpcf（G）≤6。在本文中，我们证实了没有长度为 3、5 或 6 个周期的平面图的这一猜想。