Defective DP-colorings of sparse simple graphs

TL;DR

This paper investigates defective DP-colorings, a generalization of list coloring, establishing sharp bounds on the minimum edges in critical graphs for certain parameters and infinitely many graph sizes.

Contribution

It introduces and analyzes $(i,j)$-defective DP-colorings, providing exact bounds for the minimum edges in DP-critical graphs for specified parameters.

Findings

Determined sharp bounds on $g_{DP}(i,j,n)$ for $i extgreater 3$, $j extgreater 2i+1$.

Established results for infinitely many graph sizes.

Extended understanding of defective DP-colorings in sparse graphs.

Abstract

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'ak and Postle. We introduce and study $(i,j)$-defective DP-colorings of simple graphs. Let $g_{DP}(i,j,n)$ be the minimum number of edges in an $n$-vertex DP-$(i,j)$-critical graph. In this paper we determine sharp bound on $g_{DP}(i,j,n)$ for each $i\geq3$ and $j\geq 2i+1$ for infinitely many $n$.