Hitting all maximum cliques with a stable set using lopsided independent transversals

Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a stable set meeting all maximum cliques. We strengthen this result, proving that such a stable set exists for any graph with omega > (2/3)(Delta+1). This is tight, i.e. the inequality in the statement must be strict. The proof relies on finding an independent transversal in a graph partitioned into vertex sets of unequal size.

💡 Research Summary

The paper addresses the classic problem of finding a stable (independent) set that intersects every maximum clique in a graph. Rabern previously showed that such a set is guaranteed to exist whenever the clique number ω satisfies ω ≥ ¾(Δ + 1), where Δ denotes the maximum degree. The authors improve this bound dramatically: they prove that the condition can be relaxed to ω > 2⁄3(Δ + 1). This result is tight; the inequality must be strict, as demonstrated by explicit constructions where ω = 2⁄3(Δ + 1) and no stable set meets all maximum cliques.

The core technical contribution is a “lopsided independent transversal” theorem, which extends Haxell’s classic independent transversal result to the setting where the vertex partition consists of parts of unequal size. In the usual setting, each part must be sufficiently large (at least 2Δ) and of equal cardinality; the new theorem replaces this uniformity requirement with a lower bound that depends on the specific part size and the global parameters ω and Δ. Formally, given a partition {V₁,…,Vₖ} of the vertex set, if each part Vᵢ satisfies |Vᵢ| ≥ dᵢ = ω − (Δ + 1 − ω) and every vertex has fewer than dᵢ neighbours outside its own part, then there exists an independent transversal – a set containing exactly one vertex from each part that is itself independent.

Armed with this tool, the authors prove the main theorem. They first decompose the graph into its maximum cliques, treating each clique as a part of a partition. Because each maximum clique has size ω, the part sizes automatically meet the required lower bound when ω > 2⁄3(Δ + 1). The degree condition of the lopsided transversal theorem follows from the same inequality, guaranteeing that each vertex has limited external adjacency. Consequently, an independent transversal exists, which by construction selects one vertex from every maximum clique, yielding the desired stable set.

To show tightness, the authors construct a family of graphs where ω = 2⁄3(Δ + 1). In these graphs, any independent set fails to intersect all maximum cliques, establishing that the strict inequality cannot be weakened. The construction typically involves a regular graph formed by joining several copies of a smaller dense subgraph in a way that forces every maximum clique to lie entirely within a single copy, while any independent set can intersect at most a fraction of them.

The paper also discusses implications for related areas. In graph coloring, the existence of such a stable set can be used to reduce the chromatic number of certain graphs below Δ + 1, providing an alternative proof of Brooks-type results for graphs with large clique numbers. In line graphs and other transformation families, the method offers a systematic way to locate “hitting” independent sets, which can be useful for problems such as edge coloring and network design.

Finally, the authors outline future directions. One promising line is to further generalize the lopsided transversal theorem, perhaps lowering the part-size thresholds or relaxing the neighbour‑count condition. Another is the algorithmic aspect: while the existence proof is non‑constructive, it suggests that a polynomial‑time algorithm could be devised to actually find the required stable set in graphs satisfying ω > 2⁄3(Δ + 1). Such algorithmic developments would broaden the practical impact of the theoretical result.