A subset T of vertices in a hypergraph H is a transversal if T has a nonempty intersection with every edge of H. The transversal number \(\tau (H)\) of H is the minimum size of a transversal in H. A hypergraph H is 3-uniform if every edge of H has size 3. Let H be a 3-uniform hypergraph with \(n_{_H}\) vertices and \(m_{_H}\) edges. Tuza (Discrete Math 86:117–126, 1990) and Chvátal and McDiarmid (Combinatorica 12:19–26, 1992) showed that \(4\tau (H) \le n_{_H}+ m_{_H}\). Chvátal and McDiarmid also showed that \(6\tau (H) \le 2n_{_H}+ m_{_H}\). The linear hypergraphs achieving equality in these bounds were characterized by the authors (Henning and Yeo in J Graph Theory 59:326–348, 2008; Discrete Math 313:959–966, 2013). In this paper, we show that these bounds can be improved if we impose some structural properties on H. We show that if H does not contain a subhypergraph isomorphic to the affine plane AG(2, 3) of order 3 with two vertices deleted, then \(17\tau (H) \le 5n_{_H}+ 3m_{_H}\). The total domination number \(\gamma _t(G)\) of a graph G is the minimum cardinality of a set S of vertices so that every vertex in G is adjacent to some vertex in S. It is known (Archdeacon et al. in J Graph Theory 46:207–210, 2004) that if G is a graph of order n with minimum degree at least 3, then \(\gamma _t(G) \le \frac{1}{2}n\), and that this bound is tight. As a consequence of our hypergraph results, we show that if G is a graph of order n with minimum degree at least 3 that contains no 4-cycles and no specified graph on 12 vertices as a subgraph, then \(\gamma _t(G) \le \frac{8}{17}n\).

如果T与H的每个边都具有非空交集，则超图H中的顶点子集T为横向。横向数\（\ tau蛋白（H）\）的ħ被最小在横向的尺寸ħ。甲超图ħ是3-均匀如果每个边缘ħ具有尺寸3令ħ是3一致超与\（N _ {_ H} \）的顶点和\（M _ {_ H} \）的边缘。Tuza（离散数学86：117–126，1990）和Chvátal和McDiarmid（Combinatorica 12：19–26，1992）表明\（4 \ tau（H）\ le n _ {_ H} + m _ {_ H} \）。Chvátal和McDiarmid还显示了\（6 \ tau（H）\ le 2n _ {_ H} + m _ {_ H} \）。作者描述了在这些范围内实现相等的线性超图（Henning和Yeo，J Graph Theory 59：326–348，2008； Discrete Math 313：959–966，2013）。在本文中，我们表明，如果我们在H上施加一些结构特性，则可以改善这些界限。我们表明，如果H不包含与3阶仿射平面AG（2，3）同构的亚超图，并且删除了两个顶点，则\（17 \ tau（H）\ le 5n _ {_ H} + 3m _ {_ H} \ ）。图G的总支配数\（\ gamma _t（G）\）是集合S的最小基数的顶点，因此G中的每个顶点都与S中的某个顶点相邻。众所周知（Archdeacon等人，J Graph Theory 46：207-210，2004），如果G是最小次数至少为3的n阶图 ，则\（\ gamma _t（G）\ le \ frac { 1} {2} n \），并且这个界限很严格。作为我们的超图结果的结果，我们表明，如果G是阶数为 n且最小度为至少3的图，它不包含4循环且在12个顶点上没有指定图作为子图，则\（\ gamma _t（G ）\ le \ frac {8} {17} n \）。