Enumerating branched orientable surface coverings over a non-orientable surface

The isomorphism classes of several types of graph coverings of a graph have been enumerated by many authors [M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87–97; S. Hong, J.H. Kwak, J. Lee, Regular graph coverings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85–105; J.H. Kwak, J.H. Chun, J.Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273–285; J.H. Kwak, J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. XLII (1990) 747–761; J.H. Kwak, J. Lee, Enumeration of connected graph coverings, J. Graph Theory 23 (1996) 105–109]. Recently, Kwak et al [Balanced regular coverings of a signed graph and regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004) 177–193] enumerated the isomorphism classes of balanced regular coverings of a signed graph, as a continuation of an enumeration work done by Archdeacon et al [Bipartite covering graphs, Discrete Math. 214 (2000) 51–63] the isomorphism classes of branched orientable regular surface coverings of a non-orientable surface having a finite abelian covering transformation group. In this paper, we enumerate the isomorphism classes of connected balanced (regular or irregular) coverings of a signed graph and those of unbranched orientable coverings of a non-orientable surface, as an answer of the question raised by Liskovets [Reductive enumeration under mutually orthogonal group actions, Acta-Appl. E-mail address: [email protected] (J.H. Kwak). 1 Supported by Com2MaC-KOSEF (R11-1999-054). 2 Supported by KRF(2000-015-DP0037). 0012-365X/$ see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2003.10.030 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 43 Math. 52 (1998) 91–120]. As a consequence of these two results, we also enumerate the isomorphism classes of branched orientable surface coverings of a non-orientable surface. © 2005 Elsevier B.V. All rights reserved.