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Concise proofs for adjacent vertex-distinguishing total colorings. (English) Zbl 1221.05143
Summary: Let \(G=(V,E)\) be a graph and \(f:(V \cup E)\rightarrow [k]\) be a proper total \(k\)-coloring of \(G\). We say that \(f\) is an adjacent vertex- distinguishing total coloring if for any two adjacent vertices, the set of colors appearing on the vertex and incident edges are different. We call the smallest \(k\) for which such a coloring of \(G\) exists the adjacent vertex-distinguishing total chromatic number, and denote it by \(\chi at(G)\). Here we provide short proofs for an upper bound on the adjacent vertex-distinguishing total chromatic number of graphs of maximum degree three, and the exact values of \(\chi at(G)\) when \(G\) is a complete graph or a cycle.

MSC:
05C15 Coloring of graphs and hypergraphs
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