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Adjacent vertex distinguishing total colorings of 2-degenerate graphs. (English) Zbl 1339.05141
Summary: Let $$\phi$$ be a proper total coloring of $$G$$. We use $$C_\phi(v) = \{\phi(v) \} \cup \{\phi(u v) \mid u v \in E(G) \}$$ to denote the set of colors assigned to a vertex $$v$$ and those edges incident with $$v$$. An adjacent vertex distinguishing total coloring of a graph $$G$$ is a proper total coloring of $$G$$ such that $$C_\phi(u) \neq C_\phi(v)$$ for any $$u v \in E(G)$$. The minimum number of colors required for an adjacent vertex distinguishing total coloring of $$G$$ is denoted by $$\chi_a^{\prime\prime}(G)$$. In this paper we show that if $$G$$ is a 2-degenerate graph, then $$\chi_a^{\prime\prime}(G) \leq \max \{\varDelta(G) + 2, 6 \}$$. Moreover, we also show that when $$\varDelta \geq 5$$, $$\chi_a^{\prime\prime}(G) = \Delta(G) + 2$$ if and only if $$G$$ contains two adjacent vertices of maximum degree. Our results imply the results on outerplanar graphs, $$K_4$$-minor free graphs and graphs with maximum average degree less than 3.

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