覆盖图技术及其在离散对称结构中的应用研究

覆盖图技术是近十几年来快速发展的一个新兴研究方向，属于图论、代数图论、拓扑图论和代数组合的交叉方向。覆盖图的构造和应用具有非常好的理论研究和应用价值。自2011年申请人与Conder教授首次提出通过有限生成群来构造覆盖图以来，覆盖图的构造和分类研究在国内外都有了较大的进展。本项目深化、扩展和推广已有的工作，继续研讨覆盖图的构造和应用。研究内容的主要创新点包括以下三个方面：（1）推广对称图的对称交换覆盖图的完全分类，主要的研究课题包括：分类高阶的三度、低阶的四度等对称图的对称覆盖图。（2）首次开展对称图的对称二面体群覆盖的构造方法研究，主要研究内容有：给出二面体群覆盖的构造方法；对低阶三度对称图（如完全图，完全二分图，方块图及Petersen图等）的对称二面体群覆盖给出完全分类。（3）将覆盖图技术应用到其他离散对称结构（如正则地图等）的构造和分类中。以上研究内容均处在国际、国内领先地位。

Covering techniques are known to be an useful tool in graph theory, algebraic graph theory, topological graph theory and algebraic combinatorics. The covering construction and its application have nice theoretical research value and applications to the real world (such as the computer internet design and construction). In 2011, Conder and Ma firstly used 'universal group' method and gave a new approach of constructing covering graphs. Since then classifications of symmetric covering graphs have been improved a lot, such as the covering groups are extended to abelian groups from elementary abelian groups. .In this project, we are going to improve and generalise the Conder-Ma method, apply to further construction of covering graphs. .The unique properties of this project are the following three main aspects:.Firstly, generalise the classification of arc-transitive abelian regular covering graph, and the research topics include: classifying the symmetric covers of high order 3-valent, low order 4 or 5-valent symmetric graphs. Secondly, finding a way of constructing arc-transitive dihedral regular covering graphs, and the main research topics include: the classification of small order cubic graphs such as the complete graph, the complete bipartite graph, the 3-cube graph and the Petersen graph. Finally, applying the covering techniques to the construction or classification of regular maps and regular polytopes. Such as constructing orientably regular maps of type {nk, m} for positive integer n and k,m greater than 2; prove that there exists at least one regular 4-polytope of type {a｜b｜nc } for positive integer n and integers a,b,c greater than 2.