A note on the neighbor sum distinguishing total coloring of planar graphs

Let G = (V (G), E(G)) be a graph and phi be a proper total k-coloring of G. Let f (v) denote the sum of the color on a vertex v and colors on all the edges incident with v. phi is neighbor sum distinguishing if f (u) not equal f (v) for each edge uv is an element of E(G). The smallest integer k for which such a coloring of G exists is the neighbor sum distinguishing total chromatic number and denoted by chi ''(Sigma)(G). Pilsniak and Wozniak conjectured that for any simple graph with maximum degree Delta(G), chi ''(Sigma)(G) <= Delta(G) + 3. It is known that for any simple planar graph, chi ''(Sigma)(G) <= max{Delta(G) + 3,14) and chi ''(Sigma)(G) <= max{Delta(G) + 2,16). In this paper, by using the famous Combinatorial Nullstellensatz, we show that for any simple planar graph, chi ''(Sigma)(G) <= max{Delta(G) 2,14). The bound Delta(G) + 2 is sharp. (C) 2016 Elsevier B.V. All rights reserved.