Transversal coalitions in hypergraphs

Abstract

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 $X\cup Y$ 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 $\Psi =\{V_1,V_2,\dots ,V_p\}$ such that for all $i\in \left[p\right]$, 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 $j\in \left[p\right]\setminus \left\{i\right\}$. The transversal coalition number $C_{\tau }\left(H\right)$ in H equals the maximum order of a transversal coalition partition in H. For $k\ge 2$ a hypergraph H is k-uniform if every edge of H has cardinality k. Among other results, we prove that if $k\ge 2$ and H is a k-uniform hypergraph, then $C_{\tau }\left(H\right)\le \lfloor \frac{1}{4}{k}^2\rfloor +k+1$. Further we show that for every $k\ge 2$, there exists a k-uniform hypergraph that achieves equality in this upper bound.