STUDIES ON GRAPH FACTORS

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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.