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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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##### References:
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