Vertex irregular total labeling of cubic graphs

A vertex irregular total labeling phi of a graph G is a labeling of vertices and edges of G with labels from the set {1, 2, ... , k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x in G is the sum of its label and the labels of all edges incident with a given vertex x. The minimum k for which the graph G has a vertex irregular total labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, we determine exact value of the total vertex irregularity strength of cubic graphs and a conjecture is proposed to find tvs of r-regular graphs.