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Graphs with bounded maximum average degree and their neighbor sum distinguishing total-choice numbers. (English) Zbl 1441.05078
Summary: Let $$G$$ be a graph and $$\phi : V(G) \cup E(G) \rightarrow \{1,2, 3, \ldots, k \}$$ be a $$k$$-total coloring. Let $$w(v)$$ denote the sum of color on a vertex $$v$$ and colors assigned to edges incident to $$v$$. If $$w(u) \neq w(v)$$ whenever $$u v \in E(G)$$, then $$\phi$$ is called a neighbor sum distinguishing total coloring. The smallest integer $$k$$ such that $$G$$ has a neighbor sum distinguishing $$k$$-total coloring is denoted by $$\mathrm{tndi}_{\Sigma}(G)$$. A. J. Dong and G. H. Wang [Acta Math. Sin., Engl. Ser. 30, No. 4, 703–709 (2014; Zbl 1408.05061)] obtained the results about $$\mathrm{tndi}_{\Sigma}(G)$$ depending on the value of maximum average degree. A $$k$$-assignment $$L$$ of $$G$$ is a list assignment $$L$$ of integers to vertices and edges with $$\left|L(v)\right| = k$$ for each vertex $$v$$ and $$\left|L(e)\right| = k$$ for each edge $$e$$. A total-$$L$$-coloring is a total coloring $$\phi$$ of $$G$$ such that $$\phi(v) \in L(v)$$ whenever $$v \in V(G)$$ and $$\phi(e) \in L(e)$$ whenever $$e \in E(G)$$. We state that $$G$$ has a neighbor sum distinguishing total-$$L$$-coloring if $$G$$ has a total-$$L$$-coloring such that $$w(u) \neq w(v)$$ for all $$u v \in E(G)$$. The smallest integer $$k$$ such that $$G$$ has a neighbor sum distinguishing total-$$L$$-coloring for every $$k$$-assignment $$L$$ is denoted by $$\mathrm{Ch}_{\Sigma}^{\prime\prime}(G)$$. In this paper, we strengthen results by Dong and Wang [loc. cit.] by giving analogous results for $$\mathrm{Ch}_{\Sigma}^{\prime\prime}(G)$$.
MSC:
 05C15 Coloring of graphs and hypergraphs 05C07 Vertex degrees
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