The Adjacent Vertex Distinguishing Total Chromatic Number of Graphs

Let $G=(V,E)$ be a graph and $f$:$(V\cup E)\rightarrow [k]$ be a proper total $k$-coloring of $G$.We say that $f$ is an adjacent vertex distinguishing total coloring if for any two adjacent vertices,the set of colors appearing on the vertex and incident edges are different.We call the smallest $k$ for which such a coloring of $G$ exists the adjacent vertex distinguishing total chromatic number,and denote it by $\chi_{at}(G)$. In this paper,we show that $\chi_{at}(K_{19}-E(C_4))=20$ and $\chi_{at}(K_{21}-E(C_4))=22$.