Sufficient Conditions make Graphs Edge DP-\(\varDelta \)-Colorable

In 2018, Dvořák and Postle introduced the concept of DP-coloring which is a generalization of list coloring and recently, Bernshteyn and Kostochka used this concept to give a new coloring, called edge DP-coloring. The edge DP-chromatic number of a graph G is denoted by \(\chi _{DP}'(G)\) ${\chi }_{\mathit{DP}}^{\prime }(G)$ . Note that \(\chi _{DP}'(G) \ge \varDelta \) ${\chi }_{\mathit{DP}}^{\prime }(G)\ge \Delta$ . It is interesting to find sufficient conditions of a graph G satisfying \(\chi _{DP}'(G)=\varDelta \) ${\chi }_{\mathit{DP}}^{\prime }(G)=\Delta$ .

In this paper, we give the sufficient conditions of a graph G in terms of its maximum degree and maximum average degree satisfying \(\chi '_{DP}(G)=\varDelta \) ${\chi }_{\mathit{DP}}^{\prime }(G)=\Delta$ . Consequences are inferred for planar graphs in terms of their maximum degree and girth. Moreover, we also prove that a planar graph G with maximum degree \(\varDelta \) $\Delta$ satisfying \(\chi '_{DP}(G)=\varDelta \) ${\chi }_{\mathit{DP}}^{\prime }(G)=\Delta$ .