Abstract:A planar Eulerian triangulation is a simple plane graph in which each face is a triangle and each vertex has even degree. Such objects are known to be equivalent to spherical Latin bitrades. (A Latin bitrade describes the difference between two Latin squares of the same order.) We give a classification in the near-regular case when each vertex is of degree~$4$ or $6$ (which we call a near-homogeneous spherical Latin bitrade, or NHSLB). The classification demonstrates that any NHSLB is equal to two graphs embedded in hemispheres glued at the equator, where each hemisphere belongs to one of nine possible types, each of which may be described recursively.
Keywords: planar Eulerian triangulation; Latin bitrade; Latin square
AMS Subject Classification: 05B15 05C45 05C10