Abstract:

Let H=(V,E) be a hypergraph with vertex set V and edge set E. The hypergraph H is k-uniform if every edge of H have size k. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge in 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\geq 2, let b_{k}=\sup_{H\in {\mathscr{H}}_{k}}\frac{\tau_{t}(H)}{n_{H}+m_{H}}, where {\mathscr{H}}_{k} denotes the class of all k-uniform hypergraphs containing no isolated vertices or isolated edges or multiple edges. Recently, Bujt\'as and Henning et al. proved following results: b_{2}=\frac{2}{5}, b_{3}=\frac{1}{3}, b_{4}=\frac{2}{7}. For k\geq 5, b_{k}\leq \frac{2}{7}. What's more, b_{6}\leq\frac{1}{4}, and for k\geq 7, b_{k}\leq \frac{2}{9}. In this paper, we show that b_{5}\leq\frac{4}{15} for 5-uniform hypergraphs and this improves the upper bound of b_{5}.

Key words: hypergraph, uniform hypergraph, transversal, total transversal, total domination