NEIGHBOR SUM DISTINGUISHING TOTAL CHROMATIC NUMBER OF PLANAR GRAPHS WITHOUT 5-CYCLES

For a given graph G = (V(G), E(G)), a proper total coloring phi : V(G)boolean OR E(G) -> {1, 2,..., k} is neighbor sum distinguishing if f(u) not equal f (v) for each edge uv is an element of E(G), where f (v) = Sigma(uv)is an element of E(G) phi(uv)+phi(v), v is an element of V(G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by x(Sigma)''(G). Pilsniak and Wozniak first introduced this coloring and conjectured that x(Sigma)''(G) <= Delta(G) + 3 for any graph with maximum degree Delta(G). In this paper, by using the discharging method, we prove that for any planar graph G without 5-cycles, x(Sigma)''(G) <= max{Delta(G) + 2,10}. The bound Delta(G)+ 2 is sharp. Furthermore, we get the exact value of x(Sigma)''(G) if Delta(G) >= 9.