General Vertex-Distinguishing Total Coloring of Graphs

The general vertex-distinguishing total chromatic number of a graph G is the minimum integer k, for which the vertices and edges of G are colored using k colors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value of this chromatic number of some special graphs and propose a conjecture on the upper bound of this chromatic number.