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Total domination of graphs and small transversals of hypergraphs

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Abstract

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.

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Authors and Affiliations

  1. LIRMM, 161, rue Ada, 34392, Montpellier Cedex 5, France

    Stéphan Thomassé

  2. Department of Computer Science Royal Holloway, University of London, Egham Surrey, TW20 0EX, UK

    Anders Yeo

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  1. Stéphan Thomassé
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  2. Anders Yeo
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Correspondence to Stéphan Thomassé.

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Thomassé, S., Yeo, A. Total domination of graphs and small transversals of hypergraphs. Combinatorica 27, 473–487 (2007). https://doi.org/10.1007/s00493-007-2020-3

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  • DOI: https://doi.org/10.1007/s00493-007-2020-3

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