Graph Coverings and Their Generalization

我们考虑图覆盖及其推广中的两种对象：正则覆盖；g-cyclic一张翻唱。我们在本报告中研究了三种枚举。除了n为素数的情况外，计算图G对G的自同构群GAMMA的正则n重覆盖的同构类的一般问题仍然没有解决。G的正则4重覆盖的γ -同构类的枚举是一个自然问题。G的正则4重覆盖要么是G的Z_2*Z_2-覆盖，要么是G的z_4 -覆盖。我们列举了连通图的Z_2*Z_2-覆盖的伽玛同构类。其次，对于连通对称有向图D，有限群a和g<不是> a中的一员，我们引入了D的g环a覆盖，作为图的正则覆盖的推广。在A是具有某些性质的阿贝群且g的阶为奇的情况下，我们给出了D的两个g环A-盖相对于D的自同构群GAMMA同构的一个刻画，从而我们列举了D的g环Z_<pn>-盖对任意p（>2）的i -同构类。进一步，我们计算了D的g-环z_p -盖的γ -同构类。关于连通g-环a-盖，我们给出了连通对称有向图D的g-环a-盖的i -同构类个数的分解公式，该分解公式适用于任意有限阿贝群a和任意g<不是奇数阶> a的成员。进一步，我们列举了D的g-环A-盖的i -同构类，当A是环群Z_<pn>和任意素数p（>2）的m个Z_p拷贝的直接和时。

We consider two objects in graph coverings and their generalization : regular coverings ; g-cyclic A-covers. We study three enumerations in this repot.The general problem of counting the ismorphism classes of regular n-fold coverings of a graph G with respect to a group GAMMA of automorphisms of G is still unsolved except in the case that n is prime. The enumeration of GAMMA-isomorphism classes of regular 4-fold coverings of G is a narural problem. A regular 4-fold covering of G is either a Z_2*Z_2-covering or a Z_4-covering of G.We enumerate the GAMMA-isomorphism classes of Z_2*Z_2-coverings of a connected graph.Next, for a connected symmetric digraph D,a finite group A and g<not a member of>A,we introduce a g-cyclic A-cover of D as a generalization of a regular covering of a graph. In the case that A is an abelian group with some property and the order of g is odd, we present a characterization for two g-cyclic A-covers of D to be ismorphic with respect to a group GAMMA of automorphisms of D.Thus, we enumerate the I-isomophism classes of g-cyclic Z_<pn>-covers of D for any p (>2). Furthermore, we count the GAMMA-isomophism classes of g-cyclic Z_p-covers of D.Related to connected g-cyclic a-covers, we present a decomposition formula for the number of I-isomorphism classes of g-cyclic A-covers of a connected symmetric digraph D for any finite abelian group A and any g<not a member of>A of odd order. Furthermore, we enumerate the I-isomorphism classes of g-cyclic A-covers of D,when A is the cyclic group Z_<pn> and the direct sum of m copies of Z_p for any prime number p (>2).