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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.3 Antofagasta ago. 2019

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

Articles

Rainbow neighbourhood number of graphs

Johan Kok1 

Sudev Naduvath2 

Muhammad Kamran Jamil3 

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

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

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

Abstract

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

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

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Received: April 2018; Accepted: October 2018

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License