  DP-coloring (also known as correspondence coloring) is a generalization of
list coloring developed recently by Dvorak 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 $n$.
