The Adjacent Vertex-Distinguishing Total Chromatic Number of \(1\)-Tree

Abstract

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 \(C(u) = \{f(u)\} \cup \{f(uv) | uv \in E(G)\}\). The total \(k\)-coloring \(f\) of \(G\) is called the 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 \(G\). 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 will determine the adjacent vertex-distinguishing total chromatic number of \(1\)-trees.