Arc-transitive abelian regular covers of cubic graphs
Arc-transitive abelian regular covers of cubic graphs
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DOI：
10.1016/j.jalgebra.2013.02.035
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发表时间：
2013-08
影响因子：
0.9
通讯作者：
M. Conder;Jicheng Ma
中科院分区：
文献类型：
--
作者：
M. Conder;Jicheng Ma
关于对称图的边传递覆盖或弧传递覆盖的构造问题，近年来得到了广泛的关注。在大多数情况下，该方法涉及到电压图技术，它非常适合于寻找规则覆盖，其中覆盖变换群是循环的或初等的阿贝尔的，或者更一般地，同圈的，但当覆盖群具有其他形式时就不那么容易使用了--即使它是阿贝尔的但不是同圈的。本文介绍了一种更广泛应用的不同方法。该方法采用泛群来表示基图的自同构群的作用，并利用Reidemister-Schreier理论得到了“泛覆盖群”的表示，以及一些表示理论和其它确定适当商数的方法。然后用这个方法求出K4，K3，3，立方体Q3和Petersen图的所有弧传递阿贝尔正则覆盖。续集将为希伍德图表做同样的事情。
Quite a lot of attention has been paid recently to the construction of edge- or arc-transitive covers of symmetric graphs. In most cases, the approach has involved voltage graph techniques, which are excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian, or more generally, homocyclic, but are not so easy to use when the covering group has other forms — even when it is abelian but not homocyclic. In this paper, a different approach is introduced that can be used more widely. This new approach takes a universal group for the action of the automorphism group of the base graph, and uses Reidemeister–Schreier theory to obtain a presentation for a ‘universal covering group’, and some representation theory and other methods for determining suitable quotients. This approach is then used to find all arc-transitive abelian regular covers of K4, K3,3, the cube Q3, and the Petersen graph. A sequel will do the same for the Heawood graph.