Typical circulant double coverings of a circulant graph

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. Kwak and Lee (Canad. J. Math. XLII (1990) 747) enumerated the isomorphism classes of graph bundles and those of n-fold coverings with respect to a group of automorphisms of the base graph G which fix a spanning tree. Hofmeister (Discrete Math. 98 (1991) 175) independently enumerated the isomorphism classes of n-fold graph coverings with respect to the trivial antomorphism group of a base graph G. Also, the isomorphism classes of several kinds of graph coverings of a graph G have been enumerated by Hong et al. (Discrete Math. 148 (1996) 85), Hofmeister (Discrete Math. 143 (1995) 87; SIAM J. Discrete Math. II (1998) 286), Kwak et al. (SIAM J. Discrete Math. II (1998) 273), Kwak and Lee (J. Graph Theory 23 (1996) 105) and some others. In this paper, we aim to enumerate the isomorphism classes of circulant double coverings of a connected circulant graph. The result of our study shows that no double coverings of a circulant graph of valency 3 are circulant. We also enumerate the isomorphism classes of circulant double coverings of a certain type, called a typical covering. (C) 2003 Elsevier B.V. All rights reserved.X119sciescopu

ENUMERATING TYPICAL ABELIAN PRIME-FOLD COVERINGS OF A CIRCULANT GRAPH

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. XLlI (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. (C) 2008 Elsevier B.V. All rights reserved.X112sciescopu