A note on δ^(k)-colouring of the Cartesian product of some graphs

Abstract

The chromatic number, $\chi(G)$ of a graph $G$ is the minimum number of colours used in a proper colouring of $G$. In an improper colouring, an edge $uv$ is bad if the colours assigned to the end vertices of the edge is the same. Now, if the available colours are less than that of the chromatic number of graph $G$, then colouring the graph with the available colours lead to bad edges in $G$. The number of bad edges resulting from a $\delta^{(k)}$-colouring of $G$ is denoted by $b_{k}(G)$. In this paper, we use the concept of $\delta^{(k)}$-colouring and determine the number of bad edges in Cartesian product of some graphs.