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An improved upper bound on the adjacent vertex distinguishing total chromatic number of graphs. (English) Zbl 1383.05124
Summary: An adjacent vertex distinguishing total \(k\)-coloring of a graph \(G\) is a proper total \(k\)-coloring of \(G\) such that any pair of adjacent vertices have different sets of colors. The minimum number \(k\) needed for such a total coloring of \(G\) is denoted by \(\chi_a^{\prime \prime}(G)\). In this paper we prove that \(\chi_a^{\prime \prime}(G) \leq 2 \varDelta(G) - 1\) if \(\varDelta(G) \geq 4\), and \(\chi_a^{\prime \prime}(G) \leq \lceil \frac{5 \varDelta(G) + 8}{3} \rceil\) in general. This improves a result in D. Huang et al. [ibid. 312, No. 24, 3544–3546 (2012; Zbl 1258.05037)] which states that \(\chi_a^{\prime \prime}(G) \leq 2 \varDelta(G)\) for any graph with \(\varDelta(G) \geq 3\).

MSC:
05C15 Coloring of graphs and hypergraphs
05C07 Vertex degrees
05C35 Extremal problems in graph theory
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