On \delta^(k)-colouring of Powers of Paths and Cycles

Abstract

In an improper vertex colouring of a graph, adjacent vertices are permitted to receive same colours. An edge of an improperly coloured graph is said to be a bad edge if its end vertices have the same colour. A near-proper colouring of a graph is a colouring which minimises the number of bad edges by restricting the number of colour classes that can have adjacency among their own elements. The δ(k)colouring is a near-proper colouring of G consisting of k given colours, where 1 ≤ k ≤ χ(G)− 1, which minimises the number of bad edges by permitting at most one colour class to have adjacency among the vertices in it. In this paper, we discuss the number of bad edges of powers of paths and cycles.