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## Proyecciones (Antofagasta)

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*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.38 no.3 Antofagasta Aug. 2019

#### http://dx.doi.org/10.22199/issn.0717-6279-2019-03-0030

Articles

Rainbow neighbourhood number of graphs

^{1}CHRIST (Deemed to be University), Department of Mathematics, Bangalore -560029, Karnataka, India. email: kokkiek2@tshwane.gov.za

^{2}CHRIST (Deemed to be University) , Department of Mathematics, Bangalore -560029, Karnataka, India. email: sudev.nk@christuniversity.in

^{3}Riphah International University, Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Lahore, Pakistan. Email : m.kamran.sms@gmail.com

_{
In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely the rainbow neighbourhood number and report on preliminary results thereof. The closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one coloured vertex of each colour in the chromatic colouring of a graph is called a rainbow neighbourhood. The number of rainbow neighbourhoods in a graph G is called the rainbow neighbourhood number of G, denoted by rχ(G). We also introduce the concepts of an expanded line graph of a graph G and a v-clique of v ∈ V (G). With the help of these new concepts, we also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph G.
}

**Keywords: **Colour cluster; Colour classes; Rainbow neighbourhood; Expanded line graph; v-clique

**Mathematics Subject Classification (2000): **05C07; 05C38; 05C75; 05C85

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Received: April 2018; Accepted: October 2018