TY - JOUR

T1 - Matchings in graphs and groups

AU - Jarden, Adi

AU - Levit, Vadim E.

AU - Shwartz, Robert

N1 - Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Graph

KW - Group

KW - Independent set

KW - König–Egerváry

KW - Non-commuting graph

KW - Stable set

UR - http://www.scopus.com/inward/record.url?scp=85045107029&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2018.03.051

DO - 10.1016/j.dam.2018.03.051

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AN - SCOPUS:85045107029

SN - 0166-218X

VL - 247

SP - 216

EP - 224

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -