TY - JOUR AU - Kok, Johan AU - Naduvath, Sudev AU - Jamil, Muhammad Kamran PY - 2019/08/14 Y2 - 2024/03/28 TI - Rainbow neighbourhood number of graphs JF - Proyecciones (Antofagasta, On line) JA - Proyecciones (Antofagasta, On line) VL - 38 IS - 3 SE - DO - 10.22199/issn.0717-6279-2019-03-0030 UR - https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/2833 SP - 469-484 AB - 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 rĪ‡(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. ER -