On the Fading Number of a Graph

Johan Kok; Sudev Naduvath; Eunice Gogo Mphako-Banda

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arXiv · 1711.03549

On the Fading Number of a Graph

Johan Kok·Sudev Naduvath·Eunice Gogo Mphako-Banda

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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.

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Johan Kok, Sudev Naduvath, Eunice Gogo Mphako-Banda. 2017-11-09. On the Fading Number of a Graph. https://arxiv.org/abs/1711.03549

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