### 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 language | English |
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Pages (from-to) | 123-134 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1995 |

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### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

*15*(1), 123-134. https://doi.org/10.1007/BF01294464

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

Research output: Contribution to journal › Article

*Combinatorica*, vol. 15, no. 1, pp. 123-134. https://doi.org/10.1007/BF01294464

}

TY - JOUR

T1 - Transversals of 2-intervals, a topological approach

AU - Tardos, G.

PY - 1995/3

Y1 - 1995/3

N2 - 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.

AB - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=51249166207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249166207&partnerID=8YFLogxK

U2 - 10.1007/BF01294464

DO - 10.1007/BF01294464

M3 - Article

VL - 15

SP - 123

EP - 134

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -