2-distance Vertex-distinguishing Total Coloring of Graphs

A 2-distance vertex-distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of vertices at distance 2 have distinct sets of colors. The 2-distance vertex-distinguishing total chromatic number χd2′′(G) of G is the minimum number of colors needed for a 2-distance vertex-distinguishing total coloring of G. In this paper, we determine the 2-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, Pm × Pn, and Cm × Pn. We conjecture that every simple graph G with maximum degree Δ satisfies χd2′′(G) ≤ Δ + 3.