Abstract

Let H be a 3-uniform hypergraph of order n and size m, and let T be a subset of vertices of H. The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number $\tau_s(H)$ of H is the minimum size of a strong transversal in H. We show that $7\tau_s(H)\leq4n+2m$, and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph H of order $n\geq6$ and size m that achieves equality in this bound. A set S of vertices in a graph G is a double total dominating set of G if every vertex of G is adjacent to at least two vertices in S. The minimum cardinality of a double total dominating set of G is the double total domination number $\gamma_{\times2,t}(G)$ of G. Let G be a connected graph of order n with minimum degree at least three. As an application of our hypergraph results, we show that $\gamma_{\times2,t}(G)\leq6n/7$ with equality if and only if G is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if G is not the Heawood graph, we show that $\gamma_{\times2,t}(G)\leq11n/13$, while if G is a cubic graph different from the Heawood graph, we show that $\gamma_{\times2,t}(G)\leq5n/6$, and this bound is sharp.

MSC codes

  1. 05C65
  2. 05C69

Keywords

  1. total domination
  2. transversals
  3. cubic graphs
  4. hypergraphs

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 1336 - 1355
ISSN (online): 1095-7146

History

Submitted: 12 November 2009
Accepted: 20 August 2010
Published online: 14 October 2010

MSC codes

  1. 05C65
  2. 05C69

Keywords

  1. total domination
  2. transversals
  3. cubic graphs
  4. hypergraphs

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