On the Fading Number of a Graph

TL;DR

This paper introduces the concepts of fading numbers in graphs, analyzing how many vertices can lose their color without affecting the minimum and maximum rainbow neighbourhood numbers, thus exploring stability in graph colourings.

Contribution

It defines and studies the fading numbers of a graph, a novel measure of colour stability related to rainbow neighbourhoods under fading conditions.

Findings

Fading numbers quantify colour stability in rainbow neighbourhoods.

Fading numbers do not decrease the minimum and maximum rainbow neighbourhood numbers.

The paper establishes bounds and properties of fading numbers in various graph classes.

Abstract

The closed neighbourhood $N[v]$ of a vertex $v$ of a graph $G$, consisting of at least one vertex from all colour classes with respect to a proper colouring of $G$, is called a rainbow neighbourhood in $G$. The minimum number of vertices and the maximum number of vertices which yield rainbow neighbourhoods with respect to a chromatic colouring of $G$ are called the minimum and maximum rainbow neighbourhood numbers, denoted by $r^-_\chi(G)$, $r^+_\chi(G)$ respectively. In this paper, by a colour, we mean a solid colour and by a transparent colour, we mean the fading of a solid colour. The fading numbers of a graph $G$, denoted by $f^-(G)$, $f^+(G)$ respectively, are the maximum number of vertices for which the colour may fade to transparent without a decrease in $r^-_\chi(G)$ and $r^+_\chi(G)$ respectively.