Regular graph coverings whose covering transformation groups have the isomorphism extension property

Abstract

Enumerative results are presently a major center of interest in topological graph theory, as in the work of Gross and Furst [1], Hofmeister [5,6], Kwak and Lee [9–13] and Mull et al. [15], etc. Kwak and Lee [9] enumerated the isomorphism classes of graph bundles and those of n-fold graph coverings with respect to a group of automorphisms of the base graph which fix a spanning tree. Hofmeister [6] enumerated independently the isomorphism classes of n-fold graph coverings with respect to the trivial automorphism group of the base graph. But the enumeration of isomorphism classes of regular graph coverings has not been answered completely. As its partial answers, Hofmeister enumerated the isomorphism classes of Z 2-coverings (double coverings) with respect to any group of automorphisms of the base graph, and Sato [14] did the same work for Z p -coverings (regular prime-fold coverings). With respect to the trivial automorphism group of the base graph, Hong and Kwak [8] did the same work for Z 2 ⊕ Z 2 or Z 4-coverings, and Kwak and Lee [10] did it for Z p , Z p ⊕ Z 1 ( p ≠ q primes) or Z p 2 -coverings. As an expansion of this effort, we obtain in this paper several new algebraic characterizations for isomorphic regular coverings and derive an enumerating formula for the isomorphism classes of A-coverings of a graph G with respect to any group of automorphisms of G which fix a spanning tree, when the covering transformation group A has the isomorphism extension property. By definition, it means that every isomorphism between any two isomorphic subgroups B 1 and B 2 of A can be extended to an automorphism of A. Also, we obtain complete numerical enumeration of the isomorphism classes of Z n -coverings for all n, D n -coverings for odd n (D n is the dihedral group of order 2 n) or Z p ⊕ Z p -coverings of a graph G for prime p with respect to the trivial automorphism group of G. In addition, we applied our results to a bouquet of circles.