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Neighbor sum distinguishing total colorings of planar graphs with girth at least 5. (English) Zbl 1383.05108
Summary: Let \(G= (V,E)\) be a graph and \(\phi\) be a proper \(k\)-total coloring of \(G\). For a vertex \(v\) of \(G\), let \[ f(v)= \sum_{uv\in E(G)} \phi(uv)+ \phi(v). \] The coloring \(\phi\) is neighbor sum distinguishing if \(f(u)\neq f(v)\) for each edge \(uv\in E(G)\). The smallest integer \(k\) in such a coloring of \(G\) is the neighbor sum distinguishing total chromatic number, denoted by \(\chi^{\prime\prime}_\Sigma(G)\).
By using the famous Combinatorial Nullstellensatz, we determine \(\chi^{\prime\prime}_\Sigma(G)\) for any planar graph \(G\) with girth at least 5 and \(\Delta(G)\geq 7\).

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