Transversals and domination in uniform hypergraphs

• Authors: Csilla Bujtás;Michael A. Henning;Zsolt Tuza
• Venue: European Journal of Combinatorics
• Year: 2012

Let H=(V,E) be a hypergraph with vertex set V and edge set E of order n"H=|V| and size m"H=|E|. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H. The transversal number @t(H) of H is the minimum size of a transversal in H. A dominating set in H is a subset of vertices D@?V such that for every vertex v@?V@?D there exists an edge e@?E for which v@?e and e@?D0@?. The domination number @c(H) is the minimum cardinality of a dominating set in H. A hypergraph H is k-uniform if every edge of H has size k. We establish the following relationship between the transversal number and the domination number of uniform hypergraphs. For any two nonnegative reals a and b and for every integer k=3 the equality sup"H"@?"H"""k@c(H)/(an"H+bm"H)=sup"H"@?"H"""k"""-"""1@t(H)/(an"H+(a+b)m"H) holds, where H"k denotes the class of all k-uniform hypergraphs containing no isolated vertices. As a consequence of this result, we establish upper bounds on the domination number of a k-uniform hypergraph with minimum degree at least 1. In particular, we show that if k=3, then @c(H)@?(n"H+@?k-32@?m"H)/@?3(k-1)2@? for all H@?H"k, and this bound is sharp. More generally, for k=2 and k=3 we prove that all the essential upper bounds can be written in the unified form @c(H)@?(an"H+bm"H)/(ak+b) for constants b=0 and a-b/k.