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Neighbor sum distinguishing total coloring of planar graphs without 5-cycles. (English) Zbl 1372.05072
Summary: Let $$G$$ be a graph, a proper total coloring $$\phi : V(G) \cup E(G) \rightarrow \{1, 2, \ldots, k \}$$ is called neighbor sum distinguishing if $$f(u) \neq f(v)$$ for each edge $$uv\in E(G)$$, where $$f(v) = \sum_{uv \in E(G)} \phi(uv) + \phi(v)$$, $$v \in V(G)$$. We use $$\chi_{\Sigma}^{\prime \prime}(G)$$ to denote the smallest number $$k$$ in such a coloring of $$G$$. M. Pilśniak and M. Woźniak [“On the adjacent-vertex-distinguishing index by sums in total proper colorings”, Preprint] have already conjectured that $$\chi_{\Sigma}^{\prime \prime}(G) \leq \Delta(G) + 3$$ for any simple graph with maximum degree $$\Delta(G)$$. In this paper, we prove that for any planar graph $$G$$ without 5-cycles, $$\chi_{\Sigma}^{\prime \prime}(G)\leq \max \{\Delta(G) + 3, 10\}$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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