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A note on the adjacent vertex distinguishing total chromatic number of graphs. (English) Zbl 1258.05037
Adjacent vertex distinguishing total coloring of given graph $$G$$ is a coloring $$\phi :V(G) \cup E(G) \rightarrow \{1,2,\dots,k\}$$ such that $$\phi(x) \neq \phi(y)$$ for any adjacent or incident elements $$x,y \in V(G) \cup E(G)$$ and moreover $$C_\phi(x) \neq C_\phi(y)$$ for any adjacent vertices $$x$$ and $$y$$, where $$C_\phi(x) = \{\phi(xy) \mid xy \in E(G)\} \cup \{\phi(x)\}$$. Adjacent vertex distinguishing total chromatic number $$\chi''_a(G)$$ is the smallest value of $$k$$ for which such a coloring exists. In the main theorem the authors prove that $$\chi''_a(G) \leq 2\Delta(G)$$ for all the graphs with $$\Delta \geq 3$$. It is the partial confirmation of the conjecture formulated in [Z. Zhang et al., Sci. China, Ser. A, 48, No. 3, 289–299 (2005; Zbl 1080.05036)], stating that $$\chi''_a(G) \leq \Delta(G)+3$$ for non-trivial connected graphs. This theorem generalizes the results of X. Chen [Discrete Math. 308, No. 17, 4003–4007 (2008; Zbl 1203.05052)], J. Hulgan [Discrete Math. 309, No. 8, 2548–2550 (2009; Zbl 1221.05143)], and H. Wang [J. Comb. Optim. 14, No. 1, 87–109 (2007; Zbl 1125.05043)]. It also improves the inequality $$\chi''_a(G) \leq \Delta(G)+c$$ proved in [T. Coker and K. Johannson, Discrete Math. 312, No. 17, 2741–2750 (2012; Zbl 1245.05042)] for graphs with relatively small values of $$\Delta(G)$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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##### References:
 [1] M. Behzad, Graphs and their chromatic numbers, Ph.D. Thesis, Michigan State University, 1965. [2] Chen, X., On the adjacent vertex distinguishing total coloring numbers of graphs with $$\Delta = 3$$, Discrete math., 308, 4003-4007, (2008) · Zbl 1203.05052 [3] Coker, T.; Johannson, K., The adjacent vertex distinguishing total chromatic number, Discrete math., 312, 2741-2750, (2012) · Zbl 1245.05042 [4] Huang, D.; Wang, W., Adjacent vertex distinguishing total colorings of planar graphs with large maximum degree, Sci. sin. math., 42, 151-164, (2012), (in Chinese) [5] Hulgan, J., Concise proofs for adjacent vertex-distinguishing total colorings, Discrete math., 309, 2548-2550, (2009) · Zbl 1221.05143 [6] Vizing, V., Some unsolved problems in graph theory, Uspekhi mat. nauk, 23, 117-134, (1968), (in Russian) · Zbl 0177.52301 [7] Wang, H., On the adjacent vertex distinguishing total chromatic number of the graphs with $$\Delta(G) = 3$$, J. comb. optim., 14, 87-109, (2007) · Zbl 1125.05043 [8] Zhang, Z.; Chen, X.; Li, J.; Yao, B.; Lu, X.; Wang, J., On adjacent-vertex-distinguishing total coloring of graphs, Sci. China ser. A, 48, 289-299, (2005) · Zbl 1080.05036
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