Enumerating typical abelian prime-fold coverings of a circulant graph

Abstract

Enumerating the isomorphism classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory (see [R. Feng, J.H. Kwak, J. Kim, J. Lee, Isomorphism classes of concrete graph coverings, SIAM J. Discrete Math. 11 (1998) 265–272; R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73–85; R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196–207; SIAM J. Discrete Math. 21 (2007) 548–550 (erratum); M. Hofmeister, Graph covering projections arising from finite vector spaces over finite fields, Discrete Math. 143 (1995) 87–97; M. Hofmeister, Enumeration of concrete regular covering projections, SIAM J. Discrete Math. 8 (1995) 51–61; M. Hofmeister, A note on counting connected graph covering projections, SIAM J. Discrete Math. 11 (1998) 286–292; J.H. Kwak, J. 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]). A covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. A covering p from a Cayley graph Cay ( A , X ) onto another Cay ( Q , Y ) is called typical if the map p : A → Q on the vertex sets is a group epimorphism. Recently, the isomorphism classes of connected typical circulant r -fold coverings of a circulant graph are enumerated in [R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73–85] for r = 2 and in [R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196–207; SIAM J. Discrete Math. 21 (2007) 548–550 (erratum)] for any r . As a continuation of these works, we enumerate in this paper the isomorphism classes of typical abelian prime-fold coverings of a circulant graph.