Sparse critical graphs for defective DP-colorings

An interesting generalization of list coloring is so called DP-coloring (named after Dvorak and Postle). We study (i, j)-defective DP-colorings of simple graphs. Define gDP(i, j, n) to be the minimum number of edges in an n-vertex DP -(i, j)-critical graph. We prove sharp bounds on gDP (i, j, n) for i =1, 2 and j >= 2i for infinitely many n. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).