Typical circulant double coverings of a circulant graph

Abstract

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 automorphism 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. 11 (1998) 286), Kwak et al. (SIAM J. Discrete Math. 11 (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.