Strong Transversals in Hypergraphs and Double Total Domination in Graphs

• Authors: Michael A. Henning;Anders Yeo
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2010

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.