On the Transversal Coalition in r-Uniform Hypergraphs

A transversal coalition in a hypergraph $H$ is a partition of the vertex set $U$ into two subsets $U_1$ and $U_2$ such that neither $U_1$ nor $U_2$ alone intersects every hyperedge of $H$, but their union, $U_1 \cup U_2$, intersects every hyperedge in $H$. In this work, we investigate transversal coalition partitions in \( r \)-uniform hypergraphs. Specifically, we determine the transversal coalition number of complete $r$-uniform hypergraphs, complete bipartite $r$-uniform hypergraphs, $r$-uniform stars, and complete $r$-partite $r$-uniform hypergraphs. We also investigate the transversal coalition number of $r$-uniform linear paths and cycles.