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={V₁,V₂,…,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 [Formula presented]. Further we show that for every k≥2, there exists a k-uniform hypergraph that achieves equality in this upper bound.