As a generalization of list coloring, DP-coloring of graphs was introduced by Dvořák and Postle (2018). Recently, Bernshteyn and Kostochka introduced edge DP-coloring of graphs which is naturally corresponding to the DP-coloring of their line graphs. Let ${\chi }_{DP}^{\prime }\left(G\right)$ denote the edge DP-chromatic number of a graph $G$. In this paper, we prove that if $G$ is a planar graph with maximum degree $\Delta$ and without cycles of length $k$, then (1) ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$ if either $\Delta \ge 7$ and $k=4$ or $\Delta \ge 8$ and $k=3$; (2) ${\chi }_{DP}^{\prime }\left(G\right)\le \Delta +1$ if $\Delta \ge 9$.

中文翻译：

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平面图中的边缘DP着色

作为列表着色的概括，Dvořák和Postle（2018）引入了图的DP着色。最近，Bernshteyn和Kostochka引入了图形的边缘DP着色，这自然对应于其折线图的DP着色。让${\chi }_{dP}^{\prime }（G）$ 表示图的边缘DP色数 $G$。在本文中，我们证明$G$ 是最大度数的平面图 $\Delta$ 没有长度的循环 $ķ$，然后（1） ${\chi }_{dP}^{\prime }（G）=\Delta$ 如果有 $\Delta \ge 7$ 和 $ķ=4$ 要么 $\Delta \ge 8$ 和 $ķ=3$; （2）${\chi }_{dP}^{\prime }（G）\le \Delta +\mathrm{1个}$ 如果 $\Delta \ge 9$。