a note on δ(𝑘)-colouring of the cartesian product of some graphs

the chromatic number, χ(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 δ(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.