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Neighbor sum distinguishing total choosability of planar graphs without adjacent triangles. (English) Zbl 1357.05027
Summary: A total $$k$$-coloring of $$G$$ is a mapping $$\phi : V(G) \cup E(G) \rightarrow \{1, \dots, k \}$$ such that any two adjacent or incident elements in $$V(G) \cup E(G)$$ receive different colors. Let $$f(v)$$ denote the sum of colors of the edges incident to $$v$$ and the color of $$v$$. A $$k$$-neighbor sum distinguishing total coloring of $$G$$ is a total $$k$$-coloring of $$G$$ such that for each edge $$u v \in E(G)$$, $$f(u) \neq f(v)$$. By $$\chi_\Sigma^{\prime \prime}(G)$$, we denote the smallest value $$k$$ in such a coloring of $$G$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] first introduced this coloring and conjectured that $$\chi_\Sigma^{\prime \prime}(G) \leq \operatorname{\Delta}(G) + 3$$ for any simple graph $$G$$. Let $$L_z$$ $$(z \in V \cup E)$$ be a set of lists of integer numbers, each of size $$k$$. The smallest $$k$$ for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from $$L_z$$ for each $$z \in V \cup E$$ is called the neighbor sum distinguishing total choosability of $$G$$, and denoted by $$\mathrm{ch}_\Sigma^{\prime \prime}(G)$$. In this paper, we prove that $$\mathrm{ch}_\Sigma^{\prime \prime}(G) \leq \operatorname{\Delta}(G) + 3$$ for planar graphs without adjacent triangles with $$\operatorname{\Delta}(G) \geq 8$$, which implies that the conjecture proposed by M. Pilśniak and M. Woźniak [loc. cit.] is true for these planar graphs.

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