设H=(V,E)是以V为顶点集, E为(超)边集的超图. 如果H的每条边均含有k个顶点, 则称H是k-一致超图. 超图H的点子集T称为它的一个横贯, 如果T 与H 的每条边均相交. 超图H的全横贯是指它的一个横贯T, 并且T还满足如下性质: T中每个顶点均至少有一个邻点在T中. H 的全横贯数定义为H 的最小全横贯所含顶点的数目, 记作\tau_{t}(H). 对于整数k\geq 2, 令b_{k}=\sup_{H\in{\mathscr{H}}_{k}}\frac{\tau_{t}(H)}{n_{H}+m_{H}}, 其中n_H=|V|, m_H=|E|, {\mathscr{H}}_{k} 表示无孤立点和孤立边以及多重边的k-一致超图类. 最近, Bujt\'as和Henning等证明了如下结果: b_{2}=\frac{2}{5}, b_{3}=\frac{1}{3}, b_{4}=\frac{2}{7}; 当k\geq 5 时, 有b_{k}\leq \frac{2}{7}以及b_{6}\leq \frac{1}{4}; 当k\geq 7 时, b_{k}\leq \frac{2}{9}. 证明了对5-一致超图, b_{5}\leq \frac{4}{15}, 从而改进了当k=5 时b_k的上界.

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}.