Enumerating typical circulant covering projections onto a circulant graph

Abstract

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory ( see [ S. F. Du, D. Marusic, and A. O. Waller, J. Combin. Theory Ser. B, 74 ( 1998), pp. 276 - 290], [ S. F. Du, J. H. Kwak, and M. Y. Xu, J. Combin. Theory Ser. B, 93 ( 2005), pp. 73 - 93], [ R. Feng, J. H. Kwak, J. Kim, and J. Lee, SIAM J. Discrete Math., 11 ( 1998), pp. 265 - 272], [ R. Feng. and J. H. Kwak, Discrete Math., 277 ( 2004), pp. 73 - 85], [ C. D. Godsil and A. D. Hensel, J. Combin. Theory Ser. B., 56 ( 1992), pp. 205 - 238], [ M. Hofmeister, Discrete Math., 143 ( 1995), pp. 87 - 97], [ M. Hofmeister, SIAM J. Discrete Math., 8 ( 1995), pp. 51 - 61], [ M. Hofmeister, SIAM J. Discrete Math., 11 ( 1998), pp. 286 - 292], [ J. H. Kwak, J. Chun, and J. Lee, SIAM J. Discrete Math., 11 ( 1998), pp. 273 - 285], [ J. H. Kwak and J. Lee, Canad. J. Math., 42 ( 1990), pp. 747 - 761], and [ J. H. Kwak and J. Lee, Combinatorial and Computational Mathematics: Present and Future, ( 2001), pp. 97 - 161]). A covering projection is called circulant if its covering graph is circulant. A covering projection 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 homomorphism from A onto Q. In [ R. Feng. and J. H. Kwak, Discrete Math., 277 ( 2004), pp. 73 - 85], the authors enumerated the isomorphism classes of typical circulant double covering projections onto a circulant graph. As a continuation of this work, we enumerate in this paper the isomorphism classes of those covering projections of any folding number.