Diagonal Transformations and Cycle Parities of Quadrangulations on Surfaces

A quadrangulation G on a closed surface F 2 is a simple graph embedded in F 2 so that each face of G is quadrilateral. The diagonal slide and the diagonal rotation were defined in [1] as two transformations of quadrangulations. See Fig. 1. We also call the both transformations diagonal transformations in total. If the graph obtained by a diagonal slide is not a simple graph, then we do not apply it. If two quadrangulations G1 and G2 on a closed surface F 2 can be transformed into each other by diagonal transformations, then G1 and G2 are said to be equivalent to each other. The quadrangulations on a closed surface, except for the sphere, fall into two classes, in which one is bipartite and the other is non-bipartite. Note that there does not exist a non-bipartite quadrangulation on the sphere. The author has shown the following theorem.