On the Rainbow Neighbourhood Number of Set-Graphs

TL;DR

This paper investigates the rainbow neighbourhood numbers of set-graphs, establishes that set-graphs are perfect graphs, clarifies related coloring dilemmas, and introduces new concepts like maximax independence and the $i$-max number for future research.

Contribution

It provides new results on rainbow neighbourhood numbers of set-graphs, proves their perfection, and introduces novel graph parameters and concepts for further study.

Findings

Set-graphs are perfect graphs.

Rainbow neighbourhood numbers are characterized for set-graphs.

New graph parameters like the $i$-max number are introduced.

Abstract

In this paper, we present results for the rainbow neighbourhood numbers of set-graphs. It is also shown that set-graphs are perfect graphs. The intuitive colouring dilemma in respect of the rainbow neighbourhood convention is clarified as well. Finally, the new notion of the maximax independence, maximum proper colouring of a graph and a new graph parameter called the $i$-max number of $G$ are introduced as a new research direction.