Total Transversals in Hypergraphs and Their Applications

Abstract

Let $H = (V,E)$ be a hypergraph with vertex set $V$ and edge set $E$ of order ${n_{_H}} = |V|$ and size ${m_{_H}} = |E|$. The hypergraph $H$ is $k$-uniform if every edge of $H$ has size $k$. Two vertices in $H$ are adjacent if they belong to a common edge in $H$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. A total transversal in $H$ is a transversal $T$ in $H$ with the additional property that every vertex in $T$ is adjacent to some other vertex of $T$. The total transversal number $\tau_t(H)$ of $H$ is the minimum cardinality of a total transversal in $H$. For $k \ge 2$, let $b_k = \sup_{H \in {\cal H}_k} \, {\tau_t}(H) / ({n_{_H}} + {m_{_H}})$, where ${\cal H}_k$ denotes the class of all $k$-uniform hypergraphs containing no isolated vertices or isolated edges or multiple edges. It is known that $b_2 = 2/5$, $b_3 = 1/3$, $b_4 \le 1/3$, and $b_5 \le 2/7$. In this paper, we show that $b_4 = 2/7$ and $b_6 \le 1/4$. Further, for $k \ge 7$, we show that $b_7 \le 2/9$. These results on total transversals have applications in total domination in hypergraphs. A total dominating set in $H$ is a subset of vertices $D \subseteq V$ such that every vertex in $H$ is adjacent to some vertex in $D$. The total domination number $\gamma_t(H)$ is the minimum cardinality of a total dominating set in $H$. The following relationship between the total transversal number and the total domination number of uniform hypergraphs is known: For $k \ge 3$ and $H \in {\cal H}_k$, we have ${\gamma_t}(H) \le ( \max \{ \frac{2 }{k+1}, b_{k-1} \} ) \times {n_{_H}}$. As a consequence of our results on the total transversal number, for $k \in \{2,3,4,5,6,7,8\}$ and a hypergraph $H \in {\cal H}_k$, we have ${\gamma_t}(H) \le 2{n_{_H}}/(k+1)$.