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Neighbor sum distinguishing total coloring of the Halin graph. (Chinese. English summary) Zbl 1363.05078
Summary: Let \([k]= \{1,2,\dots,k\}\), a mapping \(\phi\) is a proper \([k]\)-total coloring of a graph \(G\). Let \(f(v)\) denote the sum of the color of vertex \(v\) and the colors of the edges incident with \(v\). A \([k]\)-neighbor sum distinguishing total coloring of \(G\) is a \([k]\)-total coloring of \(G\) such that for each edge \(uv\in E(G)\), \(f(u)\neq f(v)\). Let \(\chi^{\prime\prime}_\Sigma(G)\) denote the smallest value \(k\) in such a coloring of \(G\). There was a conjecture that \(\chi^{\prime\prime}_\Sigma(G)\leq\Delta(G)+3\) for any simple graph with maximum degree \(\Delta(G)\). By using the Combinatorial Nullstellensatz, it is shown that the conjecture holds for any Halin graph.

MSC:
05C15 Coloring of graphs and hypergraphs
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