Bounds on upper transversals in hypergraphs

Abstract

A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H. For $$k \ge 1$$, if H is a hypergraph with every edge of size at least k, then a k-transversal in H is a transversal that intersects every edge of H in at least k vertices. In particular, a 1-transversal is a transversal. The upper k-transversal number $$\Upsilon _{k}(H)$$ of H is the maximum cardinality of a minimal k-transversal in H. Let H be a hypergraph with $$n_{_H}$$ vertices and $$m_{_H}$$ edges. We show that for $$r \ge 2$$ and for every integer $$k \in [r]$$, if H is r-uniform with maximum degree $$\Delta $$, then $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{k (\Delta - 1) + r} \right) n_{_H}$$ and $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{\Delta (k + 1) + r - k} \right) (n_{_H}+ m_{_H})$$, and both bounds are tight. As a special case of this result, if H is a 3-regular, 3-uniform hypergraph, then $$\Upsilon _{2}(H) \le \frac{6}{7} n_{_H}$$, and equality in this bound is achieved by the Fano plane. We also discuss a relation between upper transversals in 3-uniform hypergraphs and the famous cap set problem, and show that for every given $$\epsilon > 0$$, there exists a 3-uniform, connected, linear hypergraphs of sufficiently large order such that $$\Upsilon _{1}(H) < \epsilon \cdot n_{_H}$$.