Transversals in regular uniform hypergraphs

The transversal number (Formula presented.) of a hypergraph (Formula presented.) is the minimum number of vertices that intersect every edge of (Formula presented.). This notion of transversal is fundamental in hypergraph theory and has been studied a great deal in the literature. A hypergraph (Formula presented.) is (Formula presented.) -regular if every vertex of (Formula presented.) has degree (Formula presented.), that is, every vertex of (Formula presented.) belongs to exactly (Formula presented.) edges. Further, (Formula presented.) is (Formula presented.) -uniform if every edge of (Formula presented.) has size (Formula presented.), and so every edge of (Formula presented.) is a (Formula presented.) -element subset of (Formula presented.). For (Formula presented.) and (Formula presented.), let (Formula presented.) be the class of all (Formula presented.) -regular (Formula presented.) -uniform hypergraphs of order (Formula presented.). In this paper we study the problem posed by Tuza to determine or estimate the best possible constants (Formula presented.) (which depend only on (Formula presented.) and (Formula presented.)) for each (Formula presented.) and (Formula presented.), such that (Formula presented.) for all (Formula presented.). These constants are given by (Formula presented.), where the supremum is taken over all (Formula presented.). Tuza presented closed formulas when (Formula presented.) or (Formula presented.), and showed that (Formula presented.) for all (Formula presented.), (Formula presented.) for (Formula presented.) even, and (Formula presented.) for (Formula presented.) odd. We conjecture that (Formula presented.) for all (Formula presented.) and that (Formula presented.) for all (Formula presented.). We show that both these conjectures hold for (Formula presented.). Moreover we show, for example, that (Formula presented.) and (Formula presented.). We show that for every (Formula presented.) and for sufficiently large (Formula presented.) and (Formula presented.), the growth of (Formula presented.) is given by (Formula presented.).