Minimal covers of infinite hypergraphs
| Autor: | Dominic van der Zypen, Taras Banakh |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Discrete mathematics
Infinite set Hypergraph Mathematics::Combinatorics 020206 networking & telecommunications 0102 computer and information sciences 02 engineering and technology Characterization (mathematics) 01 natural sciences Theoretical Computer Science Combinatorics Cover (topology) Computer Science::Discrete Mathematics 010201 computation theory & mathematics 0202 electrical engineering electronic engineering information engineering Discrete Mathematics and Combinatorics Countable set Mathematics |
| Zdroj: | Discrete Mathematics. 342:3043-3046 |
| ISSN: | 0012-365X |
| DOI: | 10.1016/j.disc.2019.06.014 |
| Popis: | For a hypergraph H = ( V , E ) , a subfamily C ⊆ E is called a cover of the hypergraph if ⋃ C = ⋃ E . A cover C is called minimal if each cover D ⊆ C of the hypergraph H coincides with C . We prove that for a hypergraph H the following conditions are equivalent: (i) each countable subhypergraph of H has a minimal cover; (ii) each non-empty subhypergraph of H has a maximal edge; (iii) H contains no isomorphic copy of the hypergraph ( ω , ω ) . This characterization implies that a countable hypergraph ( V , E ) has a minimal cover if every infinite set I ⊆ V contains a finite subset F ⊆ I such that the family of edges E F ≔ { E ∈ E : F ⊆ E } is finite. Also we prove that a hypergraph ( V , E ) has a minimal cover if sup { | E | : E ∈ E } ω or for every v ∈ V the family E v ≔ { E ∈ E : v ∈ E } is finite. |
| Databáze: | OpenAIRE |
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