NOTE [a,b]-FACTORS OF GRAPHS

For integers a and b such that 0 ≤ a ≤ b, a graph G is called an [a, b]−graph if a ≤ dG(x) ≤ b for every vertex x of G and a factor F of a graph is called an [a, b]−factor if a ≤ dF (x) ≤ b for every vertex x of F. We prove the following theorems. Let 0 ≤ l ≤ k ≤ r, 0 ≤ s, 0 ≤ u and 1 ≤ t. Then an [r, r+s]−graph has a [k, k+ t]-factor if ks ≤ rt. Moreover, if (k − l)s+ (r−k)u ≤ (r−1)t, then an [r, r + s]−graph has a [k, k + t]-factor which contains a given [l, l + u]−factor. We consider finite graphs which may have loops and multiple edges.Let G be a graph with vertex set V (G) and edge set E(G). For a vertex x of a subgraph H of G, we write dH(x) for the degree of x in H. In particular, dG(x) is the degree of x. Let a and b be integers such that 0 ≤ a ≤ b. Then G is called an [a, b]−graph if a ≤ dG(x) ≤ b for all x ∈ G.Similarly, a spanning subgraph F of a graph G is called an [a, b]−factor of G if a ≤ dF (x) ≤ b for all x ∈ G. Let g and f be integer-valued functions defined on V (G) such that g(x) ≤ f(x) for all x ∈ G. Then a (g, f)−factor of G is spanning subgraph F such that g(x) ≤ dF (x) ≤ f(x) for all x ∈ G. For a subset S of V (G), we write G − S for the subgraph G obtained from G by deleting the vertices in S together with edges incident to vertices in S. If S and T are disjoint subsets of V (G), then eG(S, T) denotes the number of edges joining S an