SHORT COMMUNICATION Some New Results on the Rainbow Neighbourhood Number of Graphs Sudev Naduvath 1 • Susanth Chandoor 2 • Sunny Joseph Kalayathankal 3 • Johan Kok 4 Received: 26 January 2017 / Revised: 30 May 2017 / Accepted: 31 July 2018 Ó The National Academy of Sciences, India 2018 Abstract A rainbow neighbourhood of a graph G is the closed neighbourhood N[v] of a vertex v 2 V ðGÞ which contains at least one coloured vertex of each colour in the chromatic colouring C of G. Let G be a graph with a chromatic colouring C defined on it. The number of ver- tices in G yielding rainbow neighbourhoods is called the rainbow neighbourhood number of the graph G, denoted by r v ðGÞ. Rainbow neighbourhood number of the comple- ments and products of certain fundamental graph classes are discussed in this paper. Keywords Colour classes  Rainbow neighbourhood  Rainbow neighbourhood number Mathematics Subject Classification 05C15  05C75 For general notations and concepts in graphs and digraphs we refer to [1–4]. For further definitions in the theory of graph colouring, see [5, 6]. Unless specified otherwise, all graphs mentioned in this paper are simple, connected and undirected graphs. A colouring of a graph is an assignment of colours to its elements (vertices and/or edges) under certain restrictions. Different types of graph colourings and related parameters have been introduced in various studies on graph colour- ings. Many practical and real-life situations paved paths to different graph colouring problems. Unless specified otherwise, by a graph colouring, we mean an assignment of colours to its vertices so that no two adjacent vertices have the same colour. A colour class of a graph is the set of all its vertices having the same colour. It can be noted that the colour classes in a graph G are independent sets in G.A k-colouring of a graph G uses k colours; it thereby partitions V into k colour classes. The chromatic number vðGÞ is the minimum value of k for which G admits a k-colouring. A graph G is said to be k- colourable if vðGÞ k and is k-chromatic if vðGÞ¼ k. Unless mentioned otherwise, we follow the convention that a colouring ðc 1 ; c 2 ; c 3 ; ...; c vðGÞ Þ is used in such a way that the colour c 1 is assigned to maximum possible number of vertices, then the colour c 2 is given to maximum pos- sible number of vertices remaining uncoloured and pro- ceeding like this, at the final step, the remaining uncoloured vertices are given colour c ‘ . This convention may be called the rainbow neighbourhood convention (see [7]). A rainbow neighbourhood of a graph G, which admits a chromatic colouring C, is the closed neighbourhood N[v] of a vertex v 2 V ðGÞ which contains at least one coloured vertex of each colour in the chromatic colouring C of G (see [7]). The number of vertices in G yielding rainbow & Sudev Naduvath sudevnk@gmail.com Susanth Chandoor susanth_c@yahoo.com Sunny Joseph Kalayathankal sunnyjoseph2014@yahoo.com Johan Kok kokkiek2@tshwane.gov.za 1 Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, Karnataka 560029, India 2 Department of Mathematics, Research and Development Centre, Bharathiar University, Coimbatore, Tamil Nadu 641046, India 3 Department of Mathematics, Kuriakose Elias College Mannanam, Kottayam, Kerala 686561, India 4 Tshwane Metropolitan Police Department, City of Tshwane, South Africa 123 Natl. Acad. Sci. Lett. https://doi.org/10.1007/s40009-018-0740-0