Transversals in regular uniform hypergraphs

Abstract

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