Introduced by Dvořák and Postle, the notion of DP-coloring generalizes list coloring and helps to prove new results on list coloring. We consider 1-defective and 2-defective DP-colorings of graphs with 2 colors. For $j=1,2$, we find exact lower bounds on the number of edges in $(j,2)$-DP-critical graphs (that is, graphs that do not admit $j$-defective DP-colorings with 2 colors but whose every proper subgraph has such a coloring) with a given number of vertices. These bounds also hold for $(j,2)$-list-critical graphs, and for $j=2$ are better than previously known bounds by Havet and Sereni (2006).

中文翻译：

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稀疏图的2缺陷DP着色

由Dvořák和Postle引入的DP着色概念可以对列表着色进行概括，并有助于证明列表着色的新结果。我们考虑具有2种颜色的图的1缺陷和2缺陷DP着色。对于$Ĵ=\mathrm{1个}，2$，我们找到了的边数的精确下界 $（Ĵ，2）$- DP-临界图（也就是不承认图$Ĵ$-具有2种颜色的瑕疵DP着色，但其每个适当的子图都具有这样的着色）并具有给定的顶点数。这些界限也适用$（Ĵ，2）$-list-critical图，以及 $Ĵ=2$ 比以前由Havet和Sereni（2006）已知的边界要好。