On J -Colouring of Chithra Graphs
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On J-Colouring of Chithra Graphs Sudev Naduvath1
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Johan Kok1
Received: 15 November 2017 / Revised: 1 January 2020 / Accepted: 18 January 2020 Ó The National Academy of Sciences, India 2020
Abstract The family of Chithra graphs is a wide ranging family of graphs which includes any graph of size at least one. Chithra graphs serve as a graph theoretical model for genetic engineering techniques or for modelling natural mutation within various biological networks found in living systems. In this paper, we discuss recently introduced J-colouring of the family of Chithra graphs. Keywords Chithra graph Chromatic colouring of graphs J-colouring of graphs Rainbow neighbourhood in a graph Mathematics Subject Classification 05C15 05C38 05C75 05C85
For general notations and concepts in graphs, see [1–3]. Unless mentioned otherwise, all graphs G mentioned in this paper are simple and finite graphs. Note that the order and size of a graph G are denoted by mðGÞ ¼ n and eðGÞ ¼ p. The minimum and maximum degrees of G are, respectively, denoted by dðGÞ and DðGÞ. The degree of a vertex v 2 VðGÞ is denoted dG ðvÞ or simply by d(v), when the context is clear. An internal vertex v of graph G is a vertex for which, dðvÞ 2. We recall that C ¼ fc1 ; c2 ; c3 ; . . .; c‘ g is a set of distinct colors and ‘ being sufficiently large, a proper vertex & Sudev Naduvath [email protected] Johan Kok [email protected] 1
Department of Mathematics, CHRIST (Deemed to be University), Bangalore, Karnataka 560029, India
colouring of a graph G denoted u : VðGÞ7!C is a vertex colouring such that no two distinct adjacent vertices have the same colour. The cardinality of a minimum set of colours which allows a proper vertex colouring of G is called the chromatic number of G and is denoted vðGÞ. When a vertex colouring is considered with colours of minimum subscripts, the colouring is called a minimum parameter colouring. Unless stated otherwise, we consider minimum parameter colour sets throughout this paper. The number of times a colour ci is allocated to vertices of a graph G is denoted by hðci Þ and u : vi 7!cj is abbreviated, cðvi Þ ¼ cj . Furthermore, if cðvi Þ ¼ cj then iðvi Þ ¼ j. We shall also colour a graph in accordance with the Rainbow Neighbourhood Convention [4]. The rainbow neighbourhood convention has been introduced in [4] as follows: consider a proper minimal colouring C ¼ fc1 ; c2 ; c3 ; . . .; c‘ g of a graph G under consideration, where ‘ ¼ vðGÞ. Colour maximum possible number of vertices of G with the colour c1 , then colour the maximum possible number of remaining uncoloured vertices with colour c2 and proceeding like this, and at the final stage colour the remaining uncoloured vertices by the colour c‘ . Such a colouring may be called a v -colouring of a graph. The inverse to the convention requires the mapping cj 7!c‘ðj1Þ . Corresponding to the inverse colouring, we define i0 ðvi Þ ¼ ‘ ðj 1Þ if cðvi Þ ¼ cj . The inverse of a v -colouring is called a vþ -colouring o
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