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Neighbor sum distinguishing total choosability of 1-planar graphs with maximum degree at least 24. (English) Zbl 1455.05018
Summary: For a simple graph $$G$$, a neighbor sum distinguishing total $$k$$-coloring of $$G$$ is a mapping $$\phi$$: $$V (G) \cup E (G) \to \{1, 2, \ldots, k\}$$ such that no two adjacent or incident elements in $$V (G) \cup E (G)$$ receive the same color and $$w_\phi (u) \neq w_\phi (v)$$ for each edge $$u v \in E (G)$$, where $$w_\phi (v)$$ (or $$w_\phi (u))$$ denotes the sum of the color of $$v$$ (or $$u)$$ and the colors of all edges incident with $$v$$ (or $$u)$$. For each element $$x \in V (G) \cup E (G)$$, let $$L(x)$$ be a list of integer numbers. If, whenever we give a list assignment $$L = \{L (x) \mid | L (x) | = k, x \in V (G) \cup E (G)\}$$, there exists a neighbor sum distinguishing total $$k$$-coloring $$\phi$$ such that $$\phi (x) \in L (x)$$ for each element $$x \in V (G) \cup E (G)$$, then we say that $$\phi$$ is a list neighbor sum distinguishing total $$k$$-coloring. The smallest $$k$$ for which such a coloring exists is called the neighbor sum distinguishing total choosability of $$G$$, denoted by $$\operatorname{ch}_{\Sigma}^{\prime\prime} (G)$$. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. There is almost no result yet about $$\operatorname{ch}_{\Sigma}^{\prime\prime} (G)$$ if $$G$$ is a 1-planar graph. We prove that $$\operatorname{ch}_{\Sigma}^{\prime\prime} (G) \leq \Delta + 3$$ for every 1-planar graph $$G$$ with maximum degree $$\Delta \geq 24$$.
##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C15 Coloring of graphs and hypergraphs 05C07 Vertex degrees 05C35 Extremal problems in graph theory
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