Transversals in 5-uniform hypergraphs and total domination in graphs with minimum degree five

Abstract

In [9] Thomassé and Yeo pose the following problem: Find the minimum ck for which every k-uniform hypergraph with n vertices and n edges has a transversal of size at most ckn. A direct consequence of this result is that every graph of order n with minimum degree at least k has a total dominating set of cardinality at most ckn. It is known that c2 = 2/3, c3 = 1/2, and c4 = 3/7. Thomassé and Yeo show that 4/11 ≤ c5 and conjecture that c5 = 4/11. In this paper we show that c5 ≤ 17/44. Thus, 16/44 ≤ c5 ≤ 17/44. More generally we prove that every 5-uniform hypergraph on n vertices and m edges has a transversal with no more than (10n+7m)/44 vertices. Consequently, every graph on n vertices with minimum degree at least five has total domination number at most 17n/44.