图的可收缩性与其团复形的无圈性

On the Contractibility of Graphs and Acyclic Propertyof Their Clique Complexes

图Ｇ的团复形是一个抽象复形，它的单形是Ｇ的团，用Ｃ（Ｇ）表示。一个复形Ｋ称为无圈的如果Ｈｑ（Ｋ）＝０（ｑ＞０），Ｈ０（Ｋ）Ｊ。Ｉｖａｓｈｃｈｅｎｋｏ（１９９４）证明了如果Ｇ是可收缩的，则Ｃ（Ｇ）是无圈的。在组合拓扑讨论班上（１９９８）。谢力同教授提出上述命题的逆命题是否成立。本文我们证明当Ｃ（Ｇ）是一个锥形或Ｃ（Ｇ）的维数小于等于２时。

The clique complex of a graph G,denoted by C(G),is an abstract complex whose simplices are the cliques of G.A complex K is said to be acyclic,if Hq(K)=0(q>0) and H0(K)J.Ivashchenko (1994) proved that the clique complex of a graph G is acyclic if G is a contractible graph.In a class having discussion on Graphs and Algebraic Topology (1998),Prof. Xie Li-tong posed whether the inverse proposition holds or not of above proposition.In this paper we prove that the inverse proposition holds if C(G) is cone or the dimension of C(G) is no more than 2.