Generalisation of the rainbow neighbourhood number and 𝑘-jump colouring of a graph

Abstract

In this paper, the notions of rainbow neighbourhood and rainbow neighbourhood number of a graph are generalised and further to these generalisations, the notion of a proper 𝑘 -jump colouring of a graph is also introduced. The generalisations follow from the understanding that a closed 𝑘 neighbourhood of a vertex 𝑣 ∈ 𝑉 ( 𝐺 ) denoted, 𝑁 𝑘 [ 𝑣 ] is the set, 𝑁 𝑘 [ 𝑣 ] = { 𝑢 : 𝑑 ( 𝑣, 𝑢 ) ≤ 𝑘, 𝑘 ∈ N and 𝑘 ≤ 𝑑𝑖𝑎𝑚 ( 𝐺 ) } . If the closed 𝑘 -neighbourhood 𝑁 𝑘 [ 𝑣 ] contains at least one of each colour of the chromatic colour set, we say that 𝑣 yields a 𝑘 -rainbow neighbourhood.