On δ ( k ) -colouring of Some Wheel Related Graphs

The question on how to colour a graph G when the number of available colours to colour G is less than that of the chromatic number χ ( G ) , such that the resulting colouring gives a minimum number of edges whose end vertices have the same colour, has been a study of great interest. Such an edge whose end vertices receive the same colour is called a bad edge. In this paper, we use the concept of δ ( k ) -colouring, where 1 ≤ k ≤ χ ( G ) − 1 , which is a near proper colouring that permits a single colour class to have adjacency between the vertices in it and restricts every other colour class to be an independent set, to find the minimum number of bad edges obtained from the same for some wheel-related graphs. The minimum number of bad edges obtained from δ ( k ) -colouring of any graph G is denoted by b k ( G ) .