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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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##### References:
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