Transversals and domination in uniform hypergraphs

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 τ ( 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 ∩ D ≠ 0̸ . The domination number γ ( 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 γ ( H ) / ( a n H + b m H ) = sup H ∈ H k − 1 τ ( H ) / ( a n 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 γ ( H ) ≤ ( n H + ⌊ k − 3 2 ⌋ 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 γ ( H ) ≤ ( a n H + b m H ) / ( a k + b ) for constants b ≥ 0 and a > − b / k . ► Let H be a hypergraph with n H vertices and m H edges. ► Let H k be the class of k -uniform hypergraphs containing no isolated vertices. ► Let τ ( H ) and γ ( H ) denote the transversal number and domination number of H , respectively. ► For k ≥ 3 an integer and for reals a , b ≥ 0 , the equality sup H ∈ H k γ ( H ) / ( a n H + b m H ) = sup H ∈ H k − 1 τ ( H ) / ( a n H + ( a + b ) m H ) holds.