Spanning bipartite quadrangulations of even triangulations

A triangulation (resp., a quadrangulation ) on a surface F is a map of a loopless graph (possibly with multiple edges) on F with each face bounded by a closed walk of length 3 (resp., 4). It is easy to see that every triangulation on any surface has a spanning quadrangulation. Kündgen and Thomassen proved that every even triangulation G (ie, each vertex has even degree) on the torus has a spanning nonbipartite quadrangulation, and that if G has sufficiently large edge width, then G also has a bipartite one. In this paper, we prove that an even triangulation G on the torus admits a spanning bipartite quadrangulation if and only if G does not have K 7 as a subgraph, and moreover, we give some other results for the problem.