Sparse critical graphs for defective DP-colorings

TL;DR

This paper investigates the minimum edge counts of critical graphs under defective DP-colorings, providing sharp bounds for specific parameters and expanding understanding of DP-coloring generalizations.

Contribution

It establishes sharp bounds on the minimum edges of DP-critical graphs for certain parameters, advancing the theory of defective DP-colorings.

Findings

Sharp bounds on $g_{DP}(i,j,n)$ for $i=1,2$ and $j extgreater=2i$

Characterization of $n$-vertex DP-critical graphs with minimal edges

Extension of DP-coloring theory to defective colorings

Abstract

An interesting generalization of list coloring is so called DP-coloring (named after Dvo\v{r}\'ak and Postle). We study $(i,j)$-defective DP-colorings of simple graphs. Define $g_{DP}(i,j,n)$ to be the minimum number of edges in an $n$-vertex DP-$(i,j)$-critical graph. We prove sharp bounds on $g_{DP}(i,j,n)$ for $i=1,2$ and $j\geq 2i$ for infinitely many $n$.