Enumeration of Regular Graph Coverings Having Finite Abelian Covering Transformation Groups

Abstract

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. An enumeration of the isomorphism classes of n-fold coverings of a graph G was done by Kwak and Lee [Canad. J. Math., XLII (1990), pp. 747--761] and independently by Hofmeister [Discrete Math., 98 (1991), pp. 437--444]. An enumeration of the isomorphism classes of connected n-fold coverings of a graph G was recently done by Kwak and Lee [J. Graph Theory, 23 (1996), pp. 105--109]. But the enumeration of the isomorphism classes of regular coverings of a graph G has been done for only a few cases. In fact, the isomorphism classes of ${\cal A}$-coverings of G were enumerated when ${\cal A}$ is the cyclic group $\BZ_n$, the dihedral group $\BD_n$ (n: odd), and the direct sum of m copies of $\BZ_p$. (See [Discrete Math., 143 (1995), pp. 87--97], [J. Graph Theory, 15 (1993), pp. 621--627], and [Discrete Math., 148 (1996), pp. 85--105]). In this paper, we discuss a method to enumerate the isomorphism classes of connected ${\cal A}$-coverings of a graph G for any finite group ${\cal A}$ and derive some formulas for enumerating the isomorphism classes of regular n-fold coverings for any natural number n. In particular, we calculate the number of the isomorphism classes of ${\cal A}$-coverings of G when ${\cal A}$ is a finite abelian group or the dihedral group $\BD_n$. Our method gives partial answers to the open problems 1 and 2 in [Discrete Math., 148 (1996), pp. 85--105] and also gives a formula to calculate the number of the subgroups of a given index of any finitely generated free abelian group.