STUDIES ON GRAPH FACTORS

首先，我们研究了完全因子与匹配可拓的关系。在这项研究中，我们得到了以下结果。设k和p为非负整数，且k≡p （mod 2）。设G是p阶的图，F是G的完全因子，满足连通性k(G) > k或分量数w(G) > 2。对于F的任意分量C，令C' =C或C-{v}，根据p-lC‘l≡k （mod 2）或p-lCl≡k-1 (mod 2)，其中v是C中的某个固定顶点。如果G-C’对所有C∈F都是k因子临界的，则G是k因子临界的。我们还证明了这个主题的几个结果。我们证明了以下定理：设G是一个图，设x是一个局部2n连通的顶点。设{u，v}是v (G)-{x}中的一对顶点，使得uv【不属于】E(G), x∈N(u)∩N(v)，且N(x)∧N(u)∪N(v)∪{u,v}。那么如果G+uv是n可扩展的，那么G是n可扩展的，或者G是图的例外族F的成员。并且证明了对于（2n+1）连通图，在相同邻域条件和删除局部连通条件下，我们得到了相同的结论。

Firstly, we treated a study of a relation between complete-factors and matching extension. In this study, we had the following result.Let k and p be nonnegative integers with k≡p (mod 2). Let G be a graph of order p, F a complete-factor of G satisfying the connectivity k(G) > k or the component number w(G) > 2. For any component C of F, let C' =C or C-{v} according as p-lC'l≡ k (mod 2) or p-lCl≡ k-1 (mod 2), where v is some fixed vertex in C. If G-C' is k-factor-critical for all C∈ F, then G is k-factor-critical. We also proved a several results for this topic.We proved the following theorem: Let G be a graph and let x be a locally 2n-connected vertex. Let {u,v} be a pair of vertices in V(G)-{x} such that uv 【not a member of】E(G), x ∈ N (u) ∩ N (v), and N (x) ⊂ N(u) ∪ N (v)∪{u,v}. Then if G+uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs.And also we proved that, for (2n+1)-connected graphs, we have the same conclusion under the same neighborhood condition and deletion of locally connected condition.