Neighbor sum distinguishing total choosability of planar graphs

A total-k-coloring of a graph G is a mapping such that any two adjacent or incident elements in receive different colors. For a total-k-coloring of G, let denote the total sum of colors of the edges incident with v and the color of v. If for each edge , , then we call such a total-k-coloring neighbor sum distinguishing. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by . PilA > niak and WoA(0)niak conjectured for any simple graph with maximum degree . In this paper, we prove that for any planar graph G with maximum degree , , where is the neighbor sum distinguishing total choosability of G.