On the $D(2)$-Vertex Distinguishing Total Coloring of Graphs with $\Delta=3$

A $D(2)$-vertex-distinguishing total coloring of a graph $G$ is a proper total coloring such that no pair of vertices, within distance two, has the same set of colors, and the minimum number of colors required for such a coloring is called $D(2)$-vertex-distinguishing total chromatic number of $G$, and denoted by $\chi_{2vt}(G)$. In this paper, we prove that $\chi_{2vt}(G)\leq11$ for any graph $G$ with $\Delta(G)=3$.