关于半传递图的若干新结果

Some New Results on 1/2-transitive Graphs

我们称无向图Ｘ为半传递的，如果它的自同构群ＡｕｔＸ在Ｘ的顶点集合以及边集合上作用是传递的，但在Ｘ的有序的相邻顶点对的集合上作用非传递．本文综述了自１９９０年以来若干数学家包括作者本人在半传递图方面研究的最新结果。特别地，我们找到了具有本原自同构群的半传递图，从而肯定地回答了Ｈｏｌｔｏｎ问题；同时还证明了只存在一个４度２７阶的半传递图，解决了Ｈｏｌｔ问题。

We call an undirected graph X 1/2-transitive,if the automorphism group Aut X of X acts transitively on the vertex set and edge set,but not on the set of ordered pairs of adjacent vertices of X.In this paper we survey some new results on 1/2-transitive graphs which have been obtained since 1990 by the author and some other researchers.In particular,the Holton’s question was affirmatively answered by finding some examples of 1/2-transitive graphs with primitive automorphism groups, and the Holt’s questions were also answered by proving that the smallest 1/2-transitive graph has 27 vertices and that there is only one such graph of degree 4.