Congruence classes of monodromies of even triangulations

A triangulation is a graph on a surface where every face is triangular. It is well-known that a planar triangulation $G$ is $3$-chromatic if and only if $G$ is even, that is, all the vertices of $G$ have even degree. We focus on even triangulations of non-spherical surfaces. It is known that there is an invariant of even triangulations, called a monodromy. This concept was firstly introduced by Hutchinson et al. [J. Combin. Theory Ser. B 84 (2002) 225--239], and the congruence class of monodromies was defined by Kawarabayashi et al. [J. Combin. Theory Ser. B 99 (2009) 229--246]. For a lower genus surface $F^2$, the number of congruence classes of monodromies have already been shown, $1$ if $F^2$ is the sphere, $2$ if $F^2$ is the projective plane by Mohar [Discrete Math. 244 (2002) 339--343] and $3$ if $F^2$ is the torus by Higuchi et al. [Discrete Math. 311 (2011) 1128--1135]. In this paper, we count the number of congruence classes of monodromies of even triangulations of general closed surface $F^2$.