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) \cup E(G) $ $ \rightarrow {1, 2, . . ., k} $ is neighbor sum distinguishing if $ f(u) \ne f(v) $ for each edge $ uv \in E(G) $, where $ f(v) = \Sigma_{ uv \in E(G) } $ $ \phi (uv) + \phi (v) $, $ v \in V (G) $. The smallest integer $k$ in such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $ \chi_\Sigma^{''} (G) $. Pilśniak and Woźniak first introduced this coloring and conjectured that $ \chi_\Sigma^{''}(G) \le \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, $ \chi_\Sigma^{''} (G) \le \text{max} \{ \Delta (G)+2, 10 \} $. The bound $ \Delta (G) + 2 $ is sharp. Furthermore, we get the exact value of $ \chi_\Sigma^{''} (G) $ if $ \Delta (G) \ge 9 $.