2-弧传递图的正则覆盖:研究综述

The Regular Covers of 2-arc-transitive Graphs:A Survey

图X的一个2-弧,指的是X的3个不同的点构成的序列(v0,v1,v2),使得(v0,v1)和(v1,v2)均为弧.图X称为是2-弧传递的,如果全自同构群Aut(X)在X的所有2-弧组成的集合上的作用是传递的.如果存在从X的顶点集到Y的顶点集的一个满射p,使得p限制在每个点的邻域上是一个双射,则称X为Y的一个覆盖,且称p为从X到Y的覆盖投影.同时,称X为覆盖图而Y为基图.一条边或者一个点的纤维指的是这条边或者这个点在p下的原像集.图X的把点纤维仍然映成点纤维的自同构称为是保纤维的,所有保纤维的自同构构成的群就称为保纤维的自同构群.图X的固定每一个点纤维集合不变的所有自同构构成的群称作覆盖变换群.如果覆盖变换群在每个点纤维上的作用是正则的,则称X是Y的一个正则覆盖.综述了2-弧传递图的正则覆盖方面的最新研究结果.特别地,对此项研究中非常关键的群论方法和拓扑图论的方法做了详细的论述.

A 2-arc of X is a sequence(v0,v1,v2)of three distinct vertices such that(v0,v1)and(v1,v2)are all arcs.The graph X is said to be 2-arc-transitive if the automorphism group Aut(X)acts transitively on the set of 2-arcs of X.If there exists a surjection p from the vertex set of X to the vertex set of Y such that p restricts to the neighbourhood of each vertex of X is a bijection,then we say X is a cover of Y,and p is the covering projection from X to Y.Moreover,X is called the covering graph and Y is the base graph.The fiber of an edge or a vertex is its preimage under p.An automorphism of X which maps a fiber to a fiber is said to be fiber-preserving.The subgroup of all those automorphisms of X which fix each of the fibers setwise is called the covering transformation group.If the action of the covering transformation group on the fibers of X is regular then we say that X is a regular cover of Y.In this paper we survey some new results on the regular covers of 2-arc-transitive graphs.In particular,we introduce the key methods of group theory and topological graph theory in detail.