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

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