  A graph is said to be {\it total-colored} if all the edges and vertices of
the graph are colored. A path in a total-colored graph is a {\it total proper
path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any
two internal adjacent vertices on the path differ in color, and $(iii)$ any
internal vertex of the path differs in color from its incident edges on the
path. A total-colored graph is called {\it total-proper connected} if any two
vertices of the graph are connected by a total proper path of the graph. For a
connected graph $G$, the {\it total proper connection number} of $G$, denoted
by $tpc(G)$, is defined as the smallest number of colors required to make $G$
total-proper connected. In this paper, we study the total proper connection
number for the graph operations. We find that $3$ is the total proper
connection number for the join, the lexicographic product and the strong
product of nearly all graphs. Besides, we study three kinds of graphs with one
factor to be traceable for the Cartesian product as well as the permutation
graphs of the star and traceable graphs. The values of the total proper
connection number for these graphs are all $3$.
