Transversals and domination in uniform hypergraphs

Csilla Bujtás, Michael A. Henning, Z. Tuza

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

Let H=(V,E) be a hypergraph with vertex set V and edge set E of order nH=V and size mH=E. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H. The transversal number (H) of H is the minimum size of a transversal in H. A dominating set in H is a subset of vertices DV such that for every vertex vVD there exists an edge eE for which ve and eD. The domination number γ(H) is the minimum cardinality of a dominating set in H. A hypergraph H is k-uniform if every edge of H has size k. We establish the following relationship between the transversal number and the domination number of uniform hypergraphs. For any two nonnegative reals a and b and for every integer k3 the equality supH∈Hkγ(H)/(anH+bmH)=supH∈Hk-1(H)/(anH+(a+b)mH) holds, where Hk denotes the class of all k-uniform hypergraphs containing no isolated vertices. As a consequence of this result, we establish upper bounds on the domination number of a k-uniform hypergraph with minimum degree at least 1. In particular, we show that if k≥3, then γ(H)(nH+⌊k-3/2⌋mH)/⌊3(k-1)/2⌋ for all H∈Hk, and this bound is sharp. More generally, for k=2 and k=3 we prove that all the essential upper bounds can be written in the unified form γ(H)≤(anH+bmH)/(ak+b) for constants b0 and a>-b/k.

Original languageEnglish
Pages (from-to)62-71
Number of pages10
JournalEuropean Journal of Combinatorics
Volume33
Issue number1
DOIs
Publication statusPublished - Jan 2012

Fingerprint

Transversals
Uniform Hypergraph
Domination
Domination number
Dominating Set
Hypergraph
Upper bound
Subset
Minimum Degree
Vertex of a graph
Cardinality
Equality
Intersection
Non-negative
Denote
Integer

ASJC Scopus subject areas

  • Geometry and Topology
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

Transversals and domination in uniform hypergraphs. / Bujtás, Csilla; Henning, Michael A.; Tuza, Z.

In: European Journal of Combinatorics, Vol. 33, No. 1, 01.2012, p. 62-71.

Research output: Contribution to journalArticle

Bujtás, Csilla ; Henning, Michael A. ; Tuza, Z. / Transversals and domination in uniform hypergraphs. In: European Journal of Combinatorics. 2012 ; Vol. 33, No. 1. pp. 62-71.
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