  In this article, we consider a proper total coloring distinguishes adjacent
vertices by sums, if every two adjacent vertices have different total sum of
colors of the edges incident to the vertex and the color of the vertex.
Pilsniak and Wozniak \cite{PW} first introduced this coloring and made a
conjecture that the minimal number of colors need to have a proper total
coloring distinguishes adjacent vertices by sums is less than or equal to the
maximum degree plus $3$. We study proper total colorings distinguishing
adjacent vertices by sums of some graphs and their products. We find that these
graphs satisfy the conjecture.
