On the Transversal Coalition in r-Uniform Hypergraphs

TL;DR

This paper studies the properties of transversal coalition partitions in r-uniform hypergraphs, determining their numbers in various hypergraph classes and exploring their structural characteristics.

Contribution

It introduces the concept of transversal coalition numbers in r-uniform hypergraphs and computes these numbers for several important classes of hypergraphs.

Findings

Transversal coalition number for complete r-uniform hypergraphs

Transversal coalition number for complete bipartite r-uniform hypergraphs

Transversal coalition number for r-uniform stars, paths, and cycles

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.