Sharp Dirac's Theorem for DP-Critical Graphs

TL;DR

This paper extends Dirac's theorem to DP-colorings, providing a minimum edge count for DP-critical graphs and classifying those meeting the bound exactly, thus advancing understanding of DP-coloring critical graphs.

Contribution

It establishes a Dirac-type theorem for DP-critical graphs and classifies list-critical graphs satisfying the bound with equality.

Findings

Proves a Dirac's theorem analogue for DP-colorings.

Classifies list-critical graphs meeting Dirac's bound with equality.

Answers a question by Kostochka and Stiebitz on critical graphs.

Abstract

Correspondence coloring, or DP-coloring, is a generalization of list coloring introduced recently by Dvo\v{r}\'{a}k and Postle. In this paper we establish a version of Dirac's theorem on the minimum number of edges in critical graphs in the framework of DP-colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz on classifying list-critical graphs that satisfy Dirac's bound with equality.