Total Colouring of Highly Irregular Graph

A proper k -total colouring of a graph G is a colouring to its vertices and edges using k colours such that no two adjacent or incident elements (vertices or edges) of G may be assigned the same colour.The k is called total chromatic number of graph G if k is minimal.The symbol χT(G) is used to denoted the chromatic number.We call a simple graph G a highly irregular graph if the degree of u' is not equal to the degree of u″ for any u',u″∈N(v) and for any vertex v of G,where N(v) is the neighborhood of v.Let G be a highly irregular graph and Δ(G) is its maximum degree.We show that if Δ(G) ≥ 2,then χT(G)=Δ(G)+1.A total colouring algorithm of G is also obtained.