Extensions and applications of the Tuza-Vestergaard theorem

The transversal number τ(H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard (2002) conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then [Formula presented]. This conjecture was recently proven by the Henning and Yeo (2023) and is now called the Tuza-Vestergaard Theorem. In this paper we extend the Tuza-Vestergaard Theorem by relaxing the 3-regularity constraint and allowing bounded maximum degree 4. We present several applications of the Tuza-Vestergaard Theorem and its extension. We obtain best known upper bounds to date on the transversal number of a (general) 6-uniform hypergraph H of order n and size m. In particular, if H is a 4-regular 6-uniform hypergraph of order n, then we show that [Formula presented]. The Tuza constant c₆ is defined by [Formula presented], where the supremum is taken over the class of all 6-uniform hypergraphs H. Since 1990 the exact value of c₆ has yet to be determined. We show that [Formula presented], where [Formula presented] is conjectured to be the correct bound. Moreover we show that if G is a graph of order n with δ(G)≥6, then [Formula presented], where γ_t(G) denotes the total domination number of G and [Formula presented] is conjectured to be the correct bound. These bounds improve best known bounds to date.