A Set and Collection Lemma

Vadim E. Levit; Eugen Mandrescu

A Set and Collection Lemma

Discrete Mathematics 2011-08-26 v5 Combinatorics

Authors: Vadim E. Levit , Eugen Mandrescu

Abstract

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.

Cite

@article{arxiv.1101.4564,
  title  = {A Set and Collection Lemma},
  author = {Vadim E. Levit and Eugen Mandrescu},
  journal= {arXiv preprint arXiv:1101.4564},
  year   = {2011}
}