$\delta^{(k)}$-Colouring of Cycle Related Graphs

With respect to a proper colouring of a graph $G$, we know that $\delta(G) \leq \chi(G) \leq \Delta(G)+1$. If distinct colours represent distinct technology types to be located at vertices the question arises on how to place at least one of each of $k$, $1\leq k < \chi(G)$ technology types together with the minimum adjacency between similar technology types. In an improper colouring an edge $uv$ such that $c(u)=c(v)$ is called a bad edge. In this paper, we introduce the notion of $\delta^{(k)}$-colouring which is a near proper colouring of $G$ with exactly $1\leq k < \chi(G)$ distinct colours which minimizes the number of bad edges.