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Upper bounds on adjacent vertex distinguishing total chromatic number of graphs. (English) Zbl 1372.05073
Summary: An adjacent vertex distinguishing total coloring of a graph $$G$$ is a proper total coloring of $$G$$ such that for any pair of adjacent vertices, the set of colors appearing on the vertex and incident edges are different. The minimum number of colors required for an adjacent vertex distinguishing total coloring of $$G$$ is denoted by $$\chi_a^{\prime\prime}(G)$$. Let $$G$$ be a graph, and $$\chi(G)$$ and $$\chi^\prime(G)$$ be the chromatic number and edge chromatic number of $$G$$, respectively. In this paper we show that $$\chi_a^{\prime\prime}(G)\leq \chi(G)+\chi^\prime(G)-1$$ for any graph $$G$$ with $$\chi(G)\geq 6$$, and $$\chi_a^{\prime\prime}(G)\leq \chi(G)+\varDelta(G)$$ for any graph $$G$$. Our results improve the only known upper bound $$2\varDelta$$ obtained by D. Huang et al. [Discrete Math. 312, No. 24, 3544–3546 (2012; Zbl 1258.05037)]. As a direct consequence, we have $$\chi_a^{\prime\prime}(G)\leq \varDelta(G)+3$$ if $$\chi(G)=3$$ and thus it implies the known results on graphs with maximum degree 3, $$K_4$$-minor-free graphs, outerplanar graphs, graphs with maximum average degree less than 3, planar graphs with girth at least 4 and 2-degenerate graphs.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C07 Vertex degrees 05C35 Extremal problems in graph theory 05C10 Planar graphs; geometric and topological aspects of graph theory
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