Matchings in graphs and groups

Berge's Lemma says that for each independent set S and maximum independent set X, there is a matching from S−X into X−S, namely, a function of S−X into X−S such that (s,f(s)) is an edge for each s∈S−X. Levit and Mandrescu prove A Set and Collection Lemma. It is a strengthening of Berge's Lemma, by which one can obtain a matching M:S−⋂Γ→⋃Γ−S, where S is an independent set and Γ is a collection of maximum independent sets. Jarden, Levit and Mandrescu invoke new inequalities from A Set and Collection Lemma and yield a new characterization of König–Egerváry graphs. In the current paper, we study Berge systems (collections of vertex sets, where the conclusion of Berge's Lemma hold). The work is divided into two parts: the general theory of Berge systems and Berge systems associated with groups (or more precisely, in non-commuting graphs). We first show that there exists many Berge systems. The notion of an ideal is central in several branches of mathematics. We define ‘a graph ideal’ which is a natural generalization of ‘an ideal’ in the context of graph theory. We show that every graph ideal carries a Berge system. We prove a sufficient condition for a Berge system to be a multi-matching system (collections where the conclusion of a Set and Collection Lemma holds). We get new inequalities in multi-matching systems. Here, we do a turn about from the general theory of Berge systems into the study of Berge systems in non-commuting graphs. We prove that for every group G, if A and B are maximal abelian subsets of 〈A,B〉_G and |A|≤|B|, then there is a non-commutative matching of A−B into B−A. It means that every non-commuting graph carries a Berge system. Moreover, we apply the sufficient condition to get a multi-matching system in the context of non-commuting graphs and invoke new inequalities in group theory. Surprisingly, the new results concerning groups, do not hold for rings.