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Neighbor sum distinguishing total choosability of planar graphs. (English) Zbl 1348.05082
Summary: A total-\(k\)-coloring of a graph \(G\) is a mapping \(c:V(G)\cup E(G)\to\{1,2,\dots,k\}\) such that any two adjacent or incident elements in \(V(G)\cup E(G)\) receive different colors. For a total-\(k\)-coloring of \(G\), let \(\sum_c(v)\) denote the total sum of colors of the edges incident with \(v\) and the color of \(v\). If for each edge \(uv\in E(G)\), \(\sum_c(u)\neq\sum_c(v)\), then we call such a total-\(k\)-coloring neighbor sum distinguishing. The least number \(k\) needed for such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number, denoted by \(\chi^{\prime\prime}_\Sigma(G)\). Pilśniak and Woźniak conjectured \(\chi^{\prime\prime}_\Sigma(G)\leq\Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that for any planar graph \(G\) with maximum degree \(\Delta (G)\), \(\mathrm{ch}^{\prime\prime}_\Sigma(G)\leq\max\{\Delta (G)+3,16\}\), where \(\mathrm{ch}^{\prime\prime}_\Sigma(G)\) is the neighbor sum distinguishing total choosability of \(G\).

MSC:
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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