Affine Planes and Transversals in 3-Uniform Linear Hypergraphs

Abstract

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 $\gamma _t(G) \le \frac{8}{17}n$.