Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10.

Given a simple graph G, a proper total-k-coloring :V(G)E(G){1,2,,k} is called neighbor sum distinguishing if S(u)S(v) for any two adjacent vertices u, vV(G), where S(u) is the sum of the color of u and the colors of the edges incident with u. It has been conjectured by Pilniak and Woniak that (G)+3 colors enable the existence of a neighbor sum distinguishing total coloring. The conjecture is confirmed for any graph with maximum degree at most 3 and for planar graph with maximum degree at least 11. We prove that the conjecture holds for any planar graph G with (G)=10. Moreover, for any planar graph G with (G) 11, (G)+2 colors guarantee such a total coloring, and the upper bound (G)+2 is tight.