On The Neighbor Sum Distinguishing Total Coloring Of Planar Graphs

Let c be a proper total coloring of a graph G = (V, E) with integers 1, 2, ..., k. For any vertex v is an element of V (G), let Sigma(c)(v) denote the sum of colors of the edges incident with v and the color of v. If for each edge uv is an element of E(G), Sigma(c)(u) not equal Sigma(c)(v), then such a total coloring is said to be neighbor sum distinguishing. The least k for which such a coloring of G exists is called the neighbor sum distinguishing total chromatic number and denoted by)( (G). Pilsniak and Wozniak conjectured chi('')(Sigma)(G) <= Delta(G) + 3 for any simple graph with maximum degree Delta(G). It is known that this conjecture holds for any planar graph with Delta(G) >= 13. In this paper, we prove that for any planar graph, chi('')(Sigma)(G) <= max{Delta(G) + 3, 14}. (C) 2015 Elsevier B.V. All rights reserved.