  Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph
$G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$
in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,
except the vertices $u$ and $v$, then $T_1,\ldots, T_k$ are completely
independent spanning trees in $G$. For $2k$-regular graphs which are
$2k$-connected, such as the Cartesian product of a complete graph of order
$2k-1$ and a cycle and some Cartesian products of three cycles (for $k=3$), the
maximum number of completely independent spanning trees contained in these
graphs is determined and it turns out that this maximum is not always $k$.
