Transversal coalitions in hypergraphs.

A transversal in a hypergraph H is set of vertices that intersect every edge of H. A transversal coalition in H consists of two disjoint sets of vertices X and Y of H , neither of which is a transversal but whose union X ∪ Y is a transversal in H. Such sets X and Y are said to form a transversal coalition. A transversal coalition partition in H is a vertex partition Ψ = { V 1 , V 2 , ... , V p } such that for all i ∈ [ p ] , either the set V i is a singleton set that is a transversal in H or the set V i forms a transversal coalition with another set V j for some j , where j ∈ [ p ] ∖ { i }. The transversal coalition number C τ (H) in H equals the maximum order of a transversal coalition partition in H. For k ≥ 2 a hypergraph H is k -uniform if every edge of H has cardinality k. Among other results, we prove that if k ≥ 2 and H is a k -uniform hypergraph, then C τ (H) ≤ ⌊ 1 4 k 2 ⌋ + k + 1. Further we show that for every k ≥ 2 , there exists a k -uniform hypergraph that achieves equality in this upper bound. [ABSTRACT FROM AUTHOR]