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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Abstract

KOK, Johan; NADUVATH, Sudev  and  JAMIL, Muhammad Kamran. Rainbow neighbourhood number of graphs. Proyecciones (Antofagasta) [online]. 2019, vol.38, n.3, pp.469-484. ISSN 0716-0917.  http://dx.doi.org/10.22199/issn.0717-6279-2019-03-0030.

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; 05C07; 05C38; 05C75; 05C85.

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