Equivalence Colouring of Graphs

Let G = (V, E) be a simple and undirected graph. A proper colouring of the vertices of V (G) is an assignment of colours to the vertices of G such that adjacent vertices receive different colours. A proper colouring of G induces a partition of V (G) into independent sets. The minimum cardinality of a proper colour partition of G is called the chromatic number of G and is denoted by (G). If in a proper colour partition of G, the union of any two-colour classes induces an acyclic subgraph, then the colouring is called acyclic colouring of G. {[4], [5], [6]}.