Enumeration of unsensed r-regular maps on the projective plane and the Klein bottle

Abstract

The paper is devoted to the problem of enumerating r-regular maps on two simplest non-orientable surfaces, the projective plane and the Klein bottle, up to all symmetries (so-called unsensed maps). We obtain general formulas that reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve the problem of explicitly describing all cyclic orbifolds for such surfaces. We also derive recurrence relations for rooted quotient maps on orbifolds that can be orientable or non-orientable surfaces with branch points and/or boundary components. These results enable us to obtain explicit formulas for the numbers of unsensed maps on the projective plane and the Klein bottle by the number of edges.