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Neighbor sum distinguishing total chromatic number of planar graphs without 5-cycles. (English) Zbl 1430.05044
Summary: For a given graph $$G = (V (G), E(G))$$, a proper total coloring $$\varphi : V (G) \cup E(G) \rightarrow \{1, 2, \dots, k\}$$ is neighbor sum distinguishing if $$f(u) \neq f(v)$$ for each edge $$uv \in E(G)$$, where $$f(v) = \Sigma_{ uv \in E(G)} \varphi (uv)+ \varphi (v)$$, $$v \in V (G)$$. The smallest integer $$k$$ in such a coloring of $$G$$ is the neighbor sum distinguishing total chromatic number, denoted by $$\chi^{\prime\prime}_\Sigma( G )$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] first introduced this coloring and conjectured that $$\chi^{\prime\prime}_\Sigma(G)\le \Delta ( G ) + 3$$ 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, $$\chi^{\prime\prime}_\Sigma ( G ) \le \max \left\{ \Delta ( G ) + 2,10\right\}$$. The bound $$\Delta (G) + 2$$ is sharp. Furthermore, we get the exact value of $$\chi^{\prime\prime}_\Sigma( G )$$ if $$\Delta (G) \geq 9$$.
##### MSC:
 05C15 Coloring of graphs and hypergraphs
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