The graph formulation of the union-closed sets conjecture

Henning Bruhn 1 Pierre Charbit 2 Oliver Schaudt 3 Jan Arne Telle 4
2 GANG - Networks, Graphs and Algorithms
LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications, Inria Paris-Rocquencourt
3 C&O - Equipe combinatoire et optimisation
IMJ-PRG - Institut de Mathématiques de Jussieu - Paris Rive Gauche
Abstract : The union-closed sets conjecture asserts that in a finite non-trivial union-closed family of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs. We derive that the union-closed sets conjecture holds for all union-closed families being the union-closure of sets of size at most three.
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Submitted on : Friday, January 8, 2016 - 5:02:44 PM
Last modification on : Wednesday, May 15, 2019 - 3:44:10 AM

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Henning Bruhn, Pierre Charbit, Oliver Schaudt, Jan Arne Telle. The graph formulation of the union-closed sets conjecture. European Journal of Combinatorics, Elsevier, 2015, 43, pp.210-219. ⟨10.1016/j.ejc.2014.08.030⟩. ⟨hal-01253152⟩



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