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Extremal Union-Closed Set Families
Umeå University, Faculty of Science and Technology, Department of Computing Science.
2019 (English)In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 35, no 6, p. 1495-1502Article in journal (Refereed) Published
Abstract [en]

A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed family forms a lattice. We call two union-closed families isomorphic if their corresponding lattices are isomorphic. Let F be a union-closed family and boolean OR(F is an element of F) F be the universe of F. Among all union-closed families isomorphic to F, we develop an algorithm to find one with a maximum universe, and an algorithm to find one with a minimum universe. We also study properties of these two extremal union-closed families in connection with the Union-Closed Set Conjecture. More specifically, a lower bound of sizes of sets of a minimum counterexample to the dual version of the Union-Closed Sets Conjecture is obtained.

Place, publisher, year, edition, pages
Springer Japan , 2019. Vol. 35, no 6, p. 1495-1502
Keywords [en]
Family of sets, Union-closed sets, Normal and irreducible families
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-167631DOI: 10.1007/s00373-019-02087-2ISI: 000507467000016Scopus ID: 2-s2.0-85073954319OAI: oai:DiVA.org:umu-167631DiVA, id: diva2:1391126
Available from: 2020-02-03 Created: 2020-02-03 Last updated: 2020-02-03Bibliographically approved

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