Defective DP-colorings of sparse multigraphs

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