Extension to Even Triangulations

Extension of a graph G is to construct a new graph with certain properties by adding edges to some pairs of vertices in G. In this paper, we focus on extension of a quadrangulation of a surface to even triangulations, where a quadrangulation is a map on a surface with every face quadrangular and a triangulation is even if all the vertices have even degree. Zhang and He [SIAM J. Comput. 34 (2005) 683–696] gave a formula for the exact number of distinct even triangulations extended from a given plane quadrangulation, and a lower bound of the number for the case of orientable non-spherical surfaces. They also posed the problem of finding the exact number for the latter case. In this paper, using topological methods, we improve the results by Zhang and He in the following directions: (I) Extension of quadrangulations of a non-orientable surface and (II) Complete enumeration of even triangulations extended from a given quadrangulation of a non-spherical surface. Indeed, we completely solve the problem by Zhang and He.