  In 1973, Altshuler characterized the $6$-regular triangulations on the torus
to be precisely those that are obtained from a regular triangulation of the $r
\times s$ toroidal grid where the vertices in the first and last column are
connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In
1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$
with $r, s \geq 3$. In this paper, we point out a gap in their classification
and show how it can be fixed. Combined with the classification of the
$4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this completes the
characterization of the colorability of all the $6$-regular triangulations on
the torus.

Combinatorics