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On the Transversal Coalition in r-Uniform Hypergraphs

Vrushali Shinde, Lata Kadam

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.

On the Transversal Coalition in r-Uniform Hypergraphs

Abstract

A transversal coalition in a hypergraph is a partition of the vertex set into two subsets and such that neither nor alone intersects every hyperedge of , but their union, , intersects every hyperedge in . In this work, we investigate transversal coalition partitions in -uniform hypergraphs. Specifically, we determine the transversal coalition number of complete -uniform hypergraph, complete bipartite -uniform hypergraph, -uniform stars, and complete -partite -uniform hypergraph. We also investigate the transversal coalition number of -uniform linear paths and cycles.
Paper Structure (2 sections, 8 theorems)

This paper contains 2 sections, 8 theorems.

Table of Contents

  1. Introduction
  2. Transversal Coalitions in paths and cycles

Key Result

Theorem 1

Hen1 A hypergraph $H$ has a transversal coalition partition if and only if there is no edge $e$ in $H$ with $e \neq V (H)$ that is a subset of all edges in $H$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Corollary 1
  • Remark 1
  • Theorem 2
  • proof
  • Definition 4
  • Theorem 3
  • ...and 13 more