  In 1973, Fisk proved that any $4$-coloring of a $3$-colorable triangulation
of the $2$-sphere can be obtained from any $3$-coloring by a sequence of
Kempe-changes. On the other hand, in the case where we are only allowed to
recolor a single vertex in each step, which is a special case of a
Kempe-change, there exists a $4$-coloring that cannot be obtained from any
$3$-coloring. In this paper, we present a characterization of a $4$-coloring of
a $3$-colorable triangulation of the $2$-sphere that can be obtained from a
$3$-coloring by a sequence of recoloring operations at single vertices, and a
criterion for a $3$-colorable triangulation of the $2$-sphere that all
$4$-colorings can be obtained from a $3$-coloring by such a sequence. Moreover,
our first result can be generalized to a high-dimensional case, in which
``$4$-coloring,'' ``$3$-colorable,'' and ``$2$-sphere'' above are replaced with
``$k$-coloring,'' ``$(k-1)$-colorable,'' and ``$(k-2)$-sphere'' for $k \geq 4$,
respectively. In addition, we show that the problem of deciding whether, for
given two $(k+1)$-colorings, one can be obtained from the other by such a
sequence is PSPACE-complete for any fixed $k \geq 4$. Our results above can be
rephrased as new results on the computational problems named {\sc
$k$-Recoloring} and {\sc Connectedness of $k$-Coloring Reconfiguration Graph},
which are fundamental problems in the field of combinatorial reconfiguration.
