Total domination of graphs and small transversals of hypergraphs

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. Chv´atal 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.

• Type: Journal
• Year: 2007
• Where: COMBINATORICA
• Authors: Stéphan Thomassé, Anders Yeo
• Keywords: 4-uniform Hypergraph, COMBINATORICA 2007, Total Domination Number, Transversal