Rainbow neighbourhood number of graphs

Kok, Johan; Naduvath, Sudev; Jamil, Muhammad Kamran; Kok, Johan; Naduvath, Sudev; Jamil, Muhammad Kamran

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

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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

Article copyright: © 2019 Johan Kokand Sudev Naduvath and Muhammad Kamran Jamil . This is an open access article distributed under the terms of the Creative Commons Licence, which permits unrestricted use and distribution provided the original author and source are credited

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