DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce and study $(i,j)$-defective DP-colorings of multigraphs. We concentrate on sparse multigraphs and consider $f_{DP}(i,j,n)$ — the minimum number of edges that may have an $n$-vertex $(i,j)$-critical multigraph, that is, a multigraph $G$ that has no $(i,j)$-defective DP-coloring but whose every proper subgraph has such a coloring. For every $i$ and $j$, we find linear lower bounds on $f_{DP}(i,j,n)$ that are exact for infinitely many $n$.

中文翻译：

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稀疏多重图的DP着色不良

DP着色（也称为对应着色）是Dvořák和Postle最近开发的列表着色的概括。介绍和研究$（\mathrm{一世}，Ĵ）$图的DP缺陷。我们专注于稀疏的多重图，并考虑$F_{dP}（\mathrm{一世}，Ĵ，ñ）$ -可能具有 $ñ$-顶点 $（\mathrm{一世}，Ĵ）$关键多重图，即多重图$G$ 那没有 $（\mathrm{一世}，Ĵ）$缺陷的DP着色，但其每个适当的子图都具有这种着色。对于每个$\mathrm{一世}$ 和 $Ĵ$，我们找到的线性下界 $F_{dP}（\mathrm{一世}，Ĵ，ñ）$ 对无限多个精确 $ñ$。