  A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a
vertex $k$-coloring such that every triple of $k$-colors appears on the
boundary of some face of $G$. The facial $3$-achromatic number $\psi_3(G)$ of
$G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete
$k$-coloring. This notion is an expansion of the complete coloring, that is, a
proper vertex coloring of a graph such that every pair of colors appears on the
ends of some edge.
  For two triangulations $G$ and $G'$ on a surface, $\psi_3(G)$ may not be
equal to $\psi_3(G')$ even if $G$ is isomorphic to $G'$ as graphs. Hence, it
would be interesting to see how large the difference between $\psi_3(G)$ and
$\psi_3(G')$ can be. We shall show that the upper bound for such difference in
terms of the genus of the surface.
