Reflection on rainbow neighbourhood numbers of graphs

TL;DR

This paper explores the properties of rainbow neighbourhood numbers in graphs, establishing their uniqueness and invariance, and clarifies their relation to minimum and maximum proper colourings.

Contribution

It introduces the concepts of minimum and maximum rainbow neighbourhood numbers, proving their uniqueness and invariance for any given graph.

Findings

Minimum and maximum rainbow neighbourhood numbers are unique for each graph.

The minimum rainbow neighbourhood number aligns with the traditional definition under the rainbow neighbourhood convention.

Relaxing the convention allows for the determination of the maximum rainbow neighbourhood number.

Abstract

A rainbow neighbourhood of a graph $G$ with respect to a proper colouring $\C$ of $G$ is the closed neighbourhood $N[v]$ of a vertex $v$ in $G$ such that $N[v]$ consists of vertices from all colour classes in $G$ with respect to $\C$. The number of vertices in $G$ which yield a rainbow neighbourhood of $G$ is called its rainbow neighbourhood number. In this paper, we show that all results known so far about the rainbow neighbourhood number of a graph $G$ implicitly refer to a minimum number of vertices which yield rainbow neighbourhoods in respect of the minimum proper colouring where the colours are allocated in accordance with the rainbow neighbourhood convention. Relaxing the aforesaid convention allows for determining a maximum rainbow neighbourhood number of a graph $G$. We also establish the fact that the minimum and maximum rainbow neighbourhood numbers are respectively, unique and therefore a constant for a given graph.