STUDIES ON GRAPH FACTORS

首先，我们研究了完全因子与匹配扩张之间的关系。在本研究中，我们得到如下结果：设k和p是非负整数，k≡p(Mod2)。设G是p阶图，F是G的完全因子，满足连通度k(G)&gt；k或分支数w(G)&gt；2.对F的任一分支C，设C‘=C或C-{v}，满足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是k因子临界的，则G是k因子临界的.证明了如下定理：设G是图，x是局部2n连通的顶点。设{u，v}是V(G)-{x}中的一对顶点，使得UV[不是]E(G)，x∈N(U)∩N(V)和N(X)⊂N(U)∪N(V)∪{u，v}。然后，如果G+UV是n-可扩的，则G是n-可扩的，或者G是例外图族F的一员，并且证明了对于(2n+1)-连通图，在相同的邻域条件和删除局部连通条件下，我们也得到了同样的结论。

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.