Rainbow Neighbourhoods of Graphs

TL;DR

This paper introduces the concept of rainbow neighbourhoods and the rainbow neighbourhood number in graphs, providing initial results and conditions for their existence, especially in line graphs.

Contribution

It defines the rainbow neighbourhood parameter and establishes a key condition for its existence in line graphs, advancing graph theory understanding.

Findings

Defined rainbow neighbourhood and its number

Provided necessary and sufficient condition for line graphs

Reported preliminary results on rainbow neighbourhoods

Abstract

In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely, the rainbow neighbourhood number of a graph $G$. We report on preliminary results thereof. We also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph $G$.