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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
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