7 research outputs found
Union-closed families of sets
Abstract3e use a lower bound on the number of small sets in an ideal to show that for each union-closed family of n sets there exists an element which belongs to at least of them, provided n is large enough
Union Closed Set Conjecture and Maximum Dicut in Connected Digraph
In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is {\it union closed} if it contains the union of any two sets in it. Let . A union closed family of sets is {\it separating} if for any two distinct elements in , there is a set in containing one of them, but not the other and there does not exist an element which is contained in every set of it. Note that any union closed family is a poset with set inclusion as the partial order relation. A separating union closed family is {\it irreducible} ({\it normalized}) if is the minimum (maximum, resp.) with respect to the poset structure of . In the part of dissertation related to this topic, we develop algorithms to transfer any given separating union closed family to a/an normalized/irreducible family without changing its poset structure. We also study properties of these two extremal union closed families in connection with the {\it Union Closed Sets Conjecture} of Frankl. Our result may lead to potential full proof of the union closed set conjecture and several other conjectures. The part of the dissertation related to the maximum edge cuts in connected digraphs goes as follows. In a given digraph , a set of edges is defined to be a {\it directed cut} if there is a nontrivial partition of such that consists of all the directed edges from to . The maximum size of a directed cut in a given digraph is denoted by , and we let be the set of all digraphs such that or for every vertex in . In this part of dissertation, we prove that for any connected digraph , which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in and answer their another question