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A note on the neighbor sum distinguishing total coloring of planar graphs. (English) Zbl 1345.05035
Summary: Let \(G = (V(G), E(G))\) be a graph and \(\phi\) be a proper total \(k\)-coloring of \(G\). Let \(f(v)\) denote the sum of the color on a vertex \(v\) and colors on all the edges incident with \(v\). \(\phi\) is neighbor sum distinguishing if \(f(u) \neq f(v)\) for each edge \(u v \in E(G)\). The smallest integer \(k\) for which such a coloring of \(G\) exists is the neighbor sum distinguishing total chromatic number and denoted by \(\chi_{\Sigma}^{\prime\prime}(G)\). M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that for any simple graph with maximum degree \(\Delta(G)\), \(\chi_{\Sigma}^{\prime\prime}(G) \leq \Delta(G) + 3\). It is known that for any simple planar graph, \(\chi_{\Sigma}^{\prime\prime}(G) \leq \max \{\Delta(G) + 3, 14 \}\) and \(\chi_{\Sigma}^{\prime\prime}(G) \leq \max \{\Delta(G) + 2, 16 \}\). In this paper, by using the famous Combinatorial Nullstellensatz, we show that for any simple planar graph, \(\chi_{\Sigma}^{\prime\prime}(G) \leq \max \{\Delta(G) + 2, 14 \}\). The bound \(\Delta(G) + 2\) is sharp.

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