Balanced regular coverings of a signed graph and regular branched orientable surface coverings over a non-orientable surface Original Research Article

The isomorphism classes of several types of graph coverings of a graph have been enumerated by many authors. Kwak and Lee (Canad. J. Math. XLII (1990) 747; J. Graph Theory 23 (1996) 105) enumerated the isomorphism classes of n-fold graph coverings of a graph G. Similar works for regular coverings of a graph can be found in (Discrete Math. 143 (1995) 87; J. Graph Theory 15 (1993) 621; Discrete Math. 148 (1996) 85; Math. Scand. 84 (1999) 23; SIAM J. Discrete Math. 11 (1998) 273). Recently, Archdeacon et al. (Discrete Math. 214 (2000) 51) characterized a bipartite covering of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of a non-bipartite graph. As a continuation of their study, we enumerate the isomorphism classes of regular balanced coverings of a signed graph and those of regular bipartite coverings of a graph. Jones (Math. Scand. 84 (1999) 23) enumerated the equivalence classes of the regular branched coverings of any given surface according to the degrees of branch points. This enables us to count the isomorphism classes of the regular branched coverings of a surface. But, it is not easy to derive a counting formula for the isomorphism classes of regular branched orientable surface coverings of a non-orientable surface. As an application of the result, we also enumerate the isomorphism classes of regular branched orientable surface coverings of a non-orientable surface having a finite abelian covering transformation group. It gives a partial answer for the question raised by Liskovets (Acta Appl. Math. 52 (1998) 91).