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On the adjacent vertex-distinguishing total chromatic numbers of the graphs with $$\Delta (G) = 3$$. (English) Zbl 1125.05043
Summary: Let $$G=(V(G),E(G))$$ be a simple graph and $$T (G)$$ be the set of vertices and edges of $$G$$. Let $$C$$ be a $$k$$-color set. A (proper) total $$k$$-coloring $$f$$ of $$G$$ is a function $$f: T(G)\rightarrow C$$ such that no adjacent or incident elements of $$T (G)$$ receive the same color. For any $$u\in V(G)$$, denote $$C(u)=\{f(u)\}\cup\{f(uv)\mid uv\in E(G)\}$$. The total $$k$$-coloring $$f$$ of $$G$$ is called adjacent vertex-distinguishing if $$C(u)\neq C(v)$$ for any edge $$uv\in E(G)$$. And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number $$\chi_{at}(G)$$ of $$G$$.
In this paper, we prove that $$\chi_{at}(G)\leq 6$$ for all connected graphs with maximum degree three. This is a step towards a conjecture on the adjacent vertex-distinguishing total coloring.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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##### References:
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