  In this paper, we study the feedback game on $3$-chromatic Eulerian
triangulations of surfaces. We prove that the winner of the game on every
$3$-chromatic Eulerian triangulation of a surface all of whose vertices have
degree $0$ modulo $4$ is always fixed. Moreover, we also study the case of
$3$-chromatic Eulerian triangulations of surfaces which have at least two
vertices whose degrees are $2$ modulo $4$, and in particular, we determine the
winner of the game on a concrete class of such graphs, called an octahedral
path.
