On strongly and robustly critical graphs

TL;DR

This paper introduces the concept of robustly critical graphs in the context of graph coloring, extending the notion of strong criticality to DP-coloring, and provides methods for constructing such graphs, including join operations with large cliques.

Contribution

It defines robustly critical graphs within DP-coloring and demonstrates that joins with large cliques produce such graphs, advancing the understanding of criticality in graph coloring.

Findings

Robustly critical graphs are not $(k-1)$-DP-colorable due to their chromatic number.

Joining a critical graph with a large clique yields a robustly critical graph.

The results extend the theory of critical graphs to the framework of DP-coloring.

Abstract

In extremal combinatorics, it is common to focus on structures that are minimal with respect to a certain property. In particular, critical and list-critical graphs occupy a prominent place in graph coloring theory. Stiebitz, Tuza, and Voigt introduced strongly critical graphs, i.e., graphs that are $k$-critical yet $L$-colorable with respect to every non-constant assignment $L$ of lists of size $k-1$. Here we strengthen this notion and extend it to the framework of DP-coloring (or correspondence coloring) by defining robustly $k$-critical graphs as those that are not $(k-1)$-DP-colorable, but only due to the fact that $\chi(G) = k$. We then seek general methods for constructing robustly critical graphs. Our main result is that if $G$ is a critical graph (with respect to ordinary coloring), then the join of $G$ with a sufficiently large clique is robustly critical; this is new even for strong criticality.