Adjacent Vertex Strongly Distinguishing Total Coloring of Unicyclic Graphs

An adjacent vertex strongly distinguishing totalcoloring of a graph G is a proper total-coloring such that no two adjacent vertices meet the same color set, where the color set of a vertex consists of all colors assigned on the vertex and its incident edges and neighbors. The minimum number of the required colors is called adjacent vertex strongly distinguishing total chromatic number, denoted by χast(G). In this paper, we first prove that χast(U) ≤ ∆(U) + 2 for a unicyclic graph U with ∆(U) ≥ 3. Then we completely determine the adjacent vertex strongly distinguishing total chromatic number of the unicyclic graph U with ∆(U) = 3, which further shows that the upper bound of χast(U) ≤ ∆(U) + 2 is sharp.