Total domination of graphs and small transversals of hypergraphs

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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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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Mathematics Subject Classification (2000)

  • 05C65
  • 05C69