On strongly and robustly critical graphs

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.