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 $χ(G) = k$. 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.

💡 Research Summary

The paper investigates minimality concepts in graph coloring, extending the classical notions of critical and list‑critical graphs to two stronger frameworks: strongly critical graphs and robustly critical graphs. A graph G is k‑critical if its chromatic number χ(G)=k and every proper subgraph has chromatic number <k. In the list‑coloring setting, a k‑assignment L assigns a list of k colors to each vertex; G is L‑critical if L is a “bad” assignment for G (no proper L‑coloring) but every proper subgraph becomes L‑colorable. Stiebitz, Tuza, and Voigt introduced strongly k‑critical graphs: these are k‑critical and, moreover, every bad (k‑1)-assignment is constant (the same set of k‑1 colors at every vertex). Thus the only obstruction to (k‑1)-list‑coloring is the chromatic number itself.

The authors generalize this idea to DP‑coloring (correspondence coloring), where not only the lists but also the identifications between colors on adjacent vertices may vary. A k‑fold cover H of G consists of a set L(v) of k “color‑vertices’’ for each vertex v and a matching between L(u) and L(v) for each edge uv. A cover is canonical if there exists a labeling λ:∪_v L(v)→{1,…,k} that is a bijection on each L(v) and respects adjacency exactly when the labels coincide. A graph G is robustly k‑critical if it is k‑critical and every bad (k‑1)-fold cover of G is canonical. Hence, in the DP‑setting the only reason G fails to be (k‑1)-DP‑colorable is its chromatic number.

The main contributions are three theorems that provide a general method for constructing such graphs. Theorem 1.4 shows that if G is any critical graph with m edges, then for every t ≥ 3m the join G ∪ K_t (adding a clique of size t and connecting it to all vertices of G) is strongly k‑critical. This extends a result of Ohba (large joins are chromatic‑choosable) and demonstrates that strong criticality can be achieved from any critical graph by a simple join operation.

Theorem 1.6 gives the analogous statement for vertex‑critical graphs, yielding strongly chromatic‑choosable graphs. Theorem 1.10 is the DP‑coloring analogue: if G is critical with m edges, then for every t ≥ 100 m³ the join G ∪ K_t is robustly k‑critical. The proof analyses the structure of DP‑covers, showing that any non‑canonical (k‑1)-fold cover would require a matching deficiency that cannot survive when a sufficiently large clique is added; the large clique forces any bad cover to be canonical.

The paper also revisits known families of strongly/robustly critical graphs, such as complete graphs, odd cycles, and the graphs E_{k,a,b} introduced by Stiebitz‑Tuza‑Voigt, and shows that joining any of these with a clique preserves robust criticality (Proposition 1.9). It points out that while every robustly critical graph is strongly critical, the converse is open (Problem 1.11). Further open questions include lowering the bound on t in the three theorems, and enumerative problems concerning the number of DP‑colorings of robustly critical graphs (Problem 1.12, Conjecture 1.13). In particular, the authors conjecture that for a k‑critical graph G, any large enough join G ∪ K_t has at least as many DP‑colorings as ordinary (k + t)-colorings.

Overall, the work provides a unified construction that turns any critical graph into a graph that is simultaneously critical, strongly critical, and (in the DP‑setting) robustly critical, simply by joining it with a sufficiently large complete graph. This bridges several strands of graph coloring theory, offers new examples for extremal investigations, and opens several avenues for further research on the interplay between ordinary, list, and DP colorings.