A Set and Collection Lemma

A set S ⊆ V (G) is independent if no two vertices from S are adjacent. Let α (G) stand for the cardinality of a largest independent set. In this paper we prove that if Λ is a nonempty collection of maximum independent sets of a graph G, and S is an independent set, then • there is a matching from S − ⋂ Λ into ⋃ Λ− S, and • |S|+ α(G) 6 ∣∣∣⋂Λ ∩ S∣∣∣+ ∣∣∣⋃Λ ∪ S∣∣∣. Based on these findings we provide alternative proofs for a number of well-known lemmata, such as the “Maximum Stable Set Lemma” due to Claude Berge and the “Clique Collection Lemma” due to András Hajnal.