On strongly and robustly critical graphs
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 (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.