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Neighbor sum distinguishing total choosability of graphs with $$\Delta = 3$$. (English) Zbl 1363.05087
Summary: Let $$G =(V, E)$$ be a graph and $$\phi: V\cup E\to \{1, 2,\dots, k\}$$ be a proper total coloring of $$G$$. Let $$f(v)$$ denote the sum of the color on vertex $$v$$ and the colors on the edges incident with $$v$$. We say that the proper total coloring $$\phi$$ is neighbor sum distinguishing if for each edge $$uv\in E(G)$$, $$f(u) \neq f(v)$$. Previous researchers introduced this coloring and conjectured that such coloring exists for any graph $$G$$ if $$k \geq \Delta(G)+3$$. The maximum average degree of $$G$$ is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad$$(G)$$. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that the conjecture holds for some graphs in their list versions. More precisely, we prove that if $$G$$ is a graph with $$\Delta(G) = 3$$ and mad$$(G) < \frac{44}{15}$$, then $$\text{ ch}_\Sigma^{''}(G) \leq 6$$ (where $$\text{ ch}_\Sigma^{''}(G)$$ is the neighbor sum distinguishing total choosability of $$G$$).

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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