A degree condition for the existence of k-factors with prescribed properties

We consider only finite undirected graphs without loops or multiple edges. Let G be a graph with vertex set V(G) and edge set E(G). For a vertex x ∈V(G), we write NG(v) for the set of vertices of V(G) adjacent to v, NG[v] for NG(v) ⋃{v}, and dG(v) = |NG(v)| for the degree of v in G. If S and T are disjoint subsets of V(G), then eG(S,T) denotes the number of edges that join S and T , and G− S denotes the subgraph of G obtained from G by deleting the vertices in S together with the edges incident with them. A k-factor of G is a spanning subgraph F of G such that dF(x) = k for every x ∈ V(F). If G and H are disjoint graphs, then the join and the union are denoted by G+H and G ⋃ H , respectively. Other terminology and notation not defined here can be found in [1]. The following theorems of k-factors in terms of degree conditions are known. Theorem 1.1 (Nishimura [4]). Let k be an integer such that k ≥ 3, and let G be a connected graph of order n with n ≥ 4k− 3, kn even, and minimum degree at least k. Suppose that max(dG(u),dG(v)) ≥ n/2 for each pair of nonadjacent vertices u, v of V(G). Then G has a k-factor.