On a Total Version of 1-2-3 Conjecture

A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set {1,. .. , k}. These colors can be used to distinguish adjacent vertices of G. There are many possibilities of such a distinction. In this paper we focus on the one by the full sum of colors of a vertex, i.e. the sum of the color of the vertex, the colors on its incident edges and the colors on its adjacent vertices. This way of distinguishing vertices has similar properties to the method when we only use incident edge colors and to the corresponding 1-2-3 Conjecture .