A set S is independent if no two vertices from S are adjacent. In this paper we prove that if F is a collection of maximum independent sets of a graph, then there is a matching from S-{ intersection of all members of F} into { union of all members of F}-S, for every independent set S. Based on this finding we give alternative proofs for a number of well-known lemmata, as the "Maximum Stable Set Lemma" due to Claude Berge and the "Clique Collection Lemma" due to Andr\'as Hajnal.
Categories: cs.DM math.CO