The adjacent vertex-distinguishing total chromatic number of 1-tree

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) −→ C such that no adjacent or incident elements of T (G) receive the same color. For any u ∈ V (G), denote C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}. The total k−coloring f of G is called the adjacent vertex-distinguishing if C(u) 6= C(v) for any edge uv ∈ E(G). And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number χat(G) of G. Let G be a connected graph. If there exists an vertex v ∈ 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. MSC: 05C15