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The adjacent vertex-distinguishing total chromatic number of 1-tree. (English) Zbl 1224.05191
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 $$k$$-colour set. A (proper) total $$k$$-colouring $$f$$ of $$G$$ is a function $$f\:T(G)\to C$$ such that no adjacent or incident elements of $$T(G)$$ receive the same colour. For any $$u\in V(G)$$, let $$C(u)=\{f(u)\}\cup \{f(uv)\mid uv\in E(G)\}$$. The total $$k$$-colouring $$f$$ of $$G$$ is called adjacent vertex-distinguishing if $$C(u)\neq C(v)$$ for any edge $$uv\in E(G)$$. The smallest number of colours in an adjacent vertex-distinguishing total colouring is called the adjacent vertex-distinguishing total chromatic number $$\chi _{at}(G)$$ of $$G$$. Let $$G$$ be a connected graph. If there exists a vertex $$v\in V(G)$$ such that $$G-v$$ is a tree, then $$G$$ is a 1-tree. In this paper, we determine the adjacent vertex-distinguishing total chromatic number of 1-trees.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C05 Trees