Transversals of 2-intervals, a topological approach

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25 Citations (Scopus)

Abstract

Fix two distinct parallel lines e and f. A 2-interval is the union of an interval on e and an interval on f. We study the transversal number τ (ℋ) of families of 2-intervals ℋ. This is the cardinality of the smallest set which intersects every 2-interval in ℋ. A Gyárfás and J. Lehel [6] proved that τ(ℋ)=O(υ(ℋ)2) where ν(ℋ) is the maximum number of disjoint 2-intervals in ℋ. In the present paper we prove the tight bond τ(ℋ)≤2v(ℋ). Our result has applications in the estimation of the transversal number of other types of set systems. The method we use is topological. We associate a simplicial complex K with our system of 2-intervals and prove that a given subcomplex is contractible in K unless the required transversal exists. Then we construct a cocycle of (another subcomplex of)K to prove that the subcomplex is not contractible in K. We hope that this approach will be applicable to a wider variety of combinatorial optimization problems.

Original languageEnglish
Pages (from-to)123-134
Number of pages12
JournalCombinatorica
Volume15
Issue number1
DOIs
Publication statusPublished - Mar 1995

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Transversals
Combinatorial optimization
Interval
Set Systems
Simplicial Complex
Cocycle
Combinatorial Optimization Problem
Intersect
Cardinality
Disjoint
Union
Distinct
Line

Keywords

  • Mathematics Subject Classification (1991): 05B40, 57Q05

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Mathematics(all)

Cite this

Transversals of 2-intervals, a topological approach. / Tardos, G.

In: Combinatorica, Vol. 15, No. 1, 03.1995, p. 123-134.

Research output: Contribution to journalArticle

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