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 ${\rm Cay} ({\cal A},X)$ onto another ${\rm Cay} ({\cal Q},Y)$ is called typical if the map $p: {\cal A}\rightarrow {\cal Q}$ on the vertex sets is a group homomorphism from ${\cal A}$ onto ${\cal Q}$. 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.