Rainbow neighbourhood number of graphs

• Kok, Johan ^[2] ; Naduvath, Sudev ^[2] ; Jamil, Muhammad Kamran ^[1]
  1. [1] Riphah International University

     Riphah International University

     Pakistán

     
  2. [2] CHRIST (Deemed to be University).
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 38, Nº. 3, 2019, págs. 469-484
• Idioma: inglés
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• Resumen

  • 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.