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5-一致超图的全横贯

林苡1  倪振羽1 单而芳2,*   

  1. 1. 上海大学数学系, 上海 200444;  2. 上海大学管理学院, 上海 200444; 
  • 收稿日期:2015-11-13 出版日期:2016-06-15 发布日期:2016-06-15
  • 通讯作者: 单而芳 efshan@shu.edu.cn
  • 基金资助:

    国家自然科学基金(No. 11571222)

Total transversals in 5-uniform hypergraphs

LIN Yi1  NI Zhenyu1  SHAN Erfang2,*   

  1. 1. Department of Mathematics, Shanghai University,  Shanghai 200444,  China; 2. School of Management,   Shanghai University,  Shanghai 200444,  China
  • Received:2015-11-13 Online:2016-06-15 Published:2016-06-15

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摘要：

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.

 

关键词: 超图, 一致超图, 横贯, 全横贯, 全控制集

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