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Neighbor sum distinguishing total chromatic number of planar graphs. (English) Zbl 1427.05093
Summary: Let \(G = (V(G), E(G))\) be a graph and \(\phi\) be a proper \(k\)-total coloring of \(G\). Set \(f_\phi(v) = \sum_{u v \in E(G)} \phi(u v) + \phi(v)\), for each \(v \in V(G)\). If \(f_\phi(u) \neq f_\phi (v)\) for each edge \(uv \in E(G)\), the coloring \(\phi\) is called a \(k\)-neighbor sum distinguishing total coloring of \(G\). The smallest integer \(k\) in such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number, denoted by \(\chi_\Sigma''(G)\). In this paper, by using the famous Combinatorial Nullstellensatz, we determine \(\chi_\Sigma''(G)\) for any planar graph \(G\) with \(\Delta (G) \geq 13\).

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