On the Transversal Coalition in r-Uniform Hypergraphs

Abstract

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 hypergraph, complete bipartite \( r \)-uniform hypergraph, \( r \)-uniform stars, and complete \( r \)-partite \( r \)-uniform hypergraph. We also investigate the transversal coalition number of \( r \)-uniform linear paths and cycles.