Edge coalitions in graphs

TL;DR

This paper introduces the concept of edge coalitions in graphs, characterizes their existence in various graphs and trees, and explores coalition graphs of special graph classes.

Contribution

It formally defines edge coalitions and partitions, characterizes graphs with small and large numbers of edge coalitions, and studies their properties in specific graph families.

Findings

Edge coalitions exist in some graphs and trees.

Characterization of graphs with few or many edge coalitions.

Analysis of coalition graphs for special graph classes.

Abstract

An edge coalition in a graph $G=(V,E)$ consists of two disjoint sets of edges $E_1$ and $E_2$, neither of which is an edge dominating set but whose union $E_1\cup E_2$ is an edge dominating set. An edge coalition partition in a graph $G$ of order $n=|V|$ and size $m$ is an edge partition $\pi=\{E_1,\cdots,E_k\}$ so that every set $E_i$ of $\pi$ either is a singleton edge dominating set, or is not an edge dominating set but forms an edge coalition with another set $E_j$ which is not an edge dominating set. In this paper we introduce the concept of edge coalition and show that there exists edge coalition for some graphs and trees. The graphs $G$ with small and size number of edge coalition are characterized. Finally, coalition graphs of special graphs are studied.