Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

Let $k$ be a positive integer. An adjacent vertex distinguishing (for short, AVD) total-$k$-coloring of a graph $G$ is a proper total-$k$-coloring of $G$ such that any two adjacent vertices have different color sets, where the color set of a vertex $v$ contains the color of $v$ and the colors of its incident edges. It was conjectured that any graph with maximum degree $\Delta$ has an AVD total-($\Delta +3$)-coloring. The conjecture was confirmed for any graph with maximum degree at most 4 and any planar graph with maximum degree at least 10. In this paper, we verify the conjecture for all planar graphs with maximum degree at least 9. Moreover, we prove that any planar graph with maximum degree at least 10 has an AVD total-($\Delta +2$)-coloring and the bound $\Delta +2$ is sharp.