On strongly and robustly critical graphs
Anton Bernshteyn, Hemanshu Kaul, Jeffrey A. Mudrock, Gunjan Sharma
TL;DR
The paper studies robustly critical graphs in the DP-coloring framework, extending the classical notion of strong criticality to canonical covers and the DP-coloring setting. It proves that if $G$ is a $k$-critical graph with $m$ edges, then the join $G \vee K_t$ is robustly $k$-critical for every $t \ge 3m$, offering a universal construction method. This result strengthens prior work on strong criticality and connects to chromatic-choosability phenomena such as Ohba's theorem. The findings provide a versatile approach to generate broad families of robustly critical graphs and illuminate the DP-coloring robustness landscape beyond previously known examples.
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 $χ(G) = k$. 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.