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Neighbor sum distinguishing total choosability of cubic graphs. (English) Zbl 07311129
Summary: Let $$G=(V, E)$$ be a graph and $${\mathbb{R}}$$ be the set of real numbers. For a $$k$$-list total assignment $$L$$ of $$G$$ that assigns to each member $$z\in V\cup E$$ a set $$L_z$$ of $$k$$ real numbers, a neighbor sum distinguishing (NSD) total $$L$$-coloring of $$G$$ is a mapping $$\phi :V\cup E \rightarrow{\mathbb{R}}$$ such that every member $$z\in V\cup E$$ receives a color of $$L_z$$, every pair of adjacent or incident members in $$V\cup E$$ receive different colors, and $$\sum_{z\in E_G(u)\cup \{u\}}\phi (z)\ne \sum_{z\in E_G(v)\cup \{v\}}\phi (z)$$ for each edge $$uv\in E$$, where $$E_G(v)$$ is the set of edges incident with $$v$$ in $$G$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] posed the conjecture that every graph $$G$$ with maximum degree $$\Delta$$ has an NSD total $$L$$-coloring with $$L_z=\{1,2,\dots , \Delta +3\}$$ for any $$z\in V\cup E$$, and confirmed the conjecture for all cubic graphs. In this paper, we extend their result by proving that every cubic graph has an NSD total $$L$$-coloring for any 6-list total assignment $$L$$.
##### MSC:
 05C15 Coloring of graphs and hypergraphs
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