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Asymptotically optimal neighbor sum distinguishing total colorings of graphs. (English) Zbl 1351.05083
Summary: Given a proper total \(k\)-coloring \(c : V(G) \cup E(G) \rightarrow \{1,2,\ldots,k\}\) of a graph \(G\), we define the value of a vertex \(v\) to be \(c(v) + \sum_{u v \in E(G)} c(u v)\). The smallest integer \(k\) such that \(G\) has a proper total \(k\)-coloring whose values form a proper coloring is the neighbor sum distinguishing total chromatic number of \(G\), \(\chi_\Sigma^{\prime \prime}(G)\). M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that \(\chi_\Sigma^{\prime \prime}(G) \leq \Delta(G) + 3\) for any simple graph with maximum degree \(\Delta(G)\). In this paper, we prove this bound to be asymptotically correct by showing that \(\chi_\Sigma^{\prime \prime}(G) \leq \Delta(G)(1 + o(1))\). The main idea of our argument relies on Przybyło’s proof [J. Przybyło’s, Random Struct. Algorithms 47, No. 4, 776–791 (2015; Zbl 1331.05083)] regarding neighbor sum distinguishing edge-colorings.

05C15 Coloring of graphs and hypergraphs
Full Text: DOI
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