$k$-Coalitions in Graphs

TL;DR

This paper introduces the concept of $k$-coalitions in graphs, exploring their properties, defining $k$-coalition partitions, and determining the $k$-coalition number for specific graphs and classes.

Contribution

It formalizes the notion of $k$-coalitions and $k$-coalition partitions, providing foundational properties and exact values for certain graphs.

Findings

Defined $k$-coalitions and $k$-coalition partitions.

Derived properties and bounds related to $k$-coalition numbers.

Calculated exact $2$-coalition numbers for specific graphs.

Abstract

In this paper, we propose and investigate the concept of $k$-coalitions in graphs, where $k\ge 1$ is an integer. A $k$-coalition refers to a pair of disjoint vertex sets that jointly constitute a $k$-dominating set of the graph, meaning that every vertex not in the set has at least $k$ neighbors in the set. We define a $k$-coalition partition of a graph as a vertex partition in which each set is either a $k$-dominating set with exactly $k$ members or forms a $k$-coalition with another set in the partition. The maximum number of sets in a $k$-coalition partition is called the $k$-coalition number of the graph represented by $C_k(G)$. We present fundamental findings regarding the properties of $k$-coalitions and their connections with other graph parameters. We obtain the exact values of $2$-coalition number of some specific graphs and also study graphs with large $2$-coalition number.