A note on hitting maximum and maximal cliques with a stable set

It was recently proved that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $\omega \geq \frac 23(\Delta+1)$ unless the graph is the strong product of $K_{\omega/2}$ and an odd hole. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.

💡 Research Summary

The paper investigates the existence of a stable (independent) set intersecting every maximum clique in a graph, focusing on the relationship between the clique number ω and the maximum degree Δ. Building on King’s 2011 result that any graph with ω > ⅔(Δ + 1) admits such a stable set, the authors extend the theorem to the boundary case ω ≥ ⅔(Δ + 1). They prove that, except for a very specific family of graphs—strong products of an odd cycle C_k (k odd) with a complete graph K_{ω/2}—every connected graph satisfying ω ≥ ⅔(Δ + 1) contains a stable set meeting all maximum cliques.

The proof proceeds by examining the clique graph G(𝒞), whose vertices correspond to maximum cliques and edges indicate non‑empty intersection. Two cases arise: (1) every connected component of G(𝒞) has a non‑empty common intersection; using Hajnal’s set lemma (|∩𝒞| + |∪𝒞| ≥ 2ω) together with King’s “lopsided independent transversal” theorem, the authors select one vertex from each component to form the desired stable set. (2) Some component has empty intersection. Lemma 4 shows that in this situation the graph must contain either C_k ⊠ K_{ω/2} (k ≥ 4) or a subgraph formed by a path or cycle of such strong products. When the strong product appears, the structure forces each maximum clique to share exactly ω/2 vertices with its neighbours, and the clique graph becomes a simple cycle of degree two. This forces the whole graph to be exactly C_k ⊠ K_{ω/2}, which indeed lacks a stable set intersecting all maximum cliques, establishing it as the unique exception.

In the second part of the paper the authors address a recent conjecture (Conjecture 6) proposing that for some universal constant ε > 0 every graph contains a stable set intersecting every maximal clique of size at least (1 − ε)(Δ + 1). They construct a counterexample: a graph composed of a large clique A of size kt and a collection of t disjoint 5‑cycles B_i, with edges between A_i and B_j whenever i ≠ j. The maximum degree is Δ = kt + 5t − 6, and every maximal clique has size at least (1 − ε)(Δ + 1) for suitably large k and t. However, any stable set that meets the large clique A must also intersect one of the B_i’s, and because each B_i is a 5‑cycle, any such intersection leaves two adjacent vertices uncovered, which together with the rest of A form a maximal clique not intersected by the stable set. Hence no stable set can hit all maximal cliques, disproving the conjecture.

Overall, the paper sharpens the known bound ω > ⅔(Δ + 1) to ω ≥ ⅔(Δ + 1) with a precise structural exception, and it demonstrates that extending the hitting‑set property to “large” maximal cliques is impossible in general. These results deepen our understanding of the interplay between clique structure, degree constraints, and independent transversals, and they suggest further investigation into restricted graph classes where analogous hitting‑set theorems might still hold.