[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 and T . Tutte[4] has shown that an r−regular graph (i.e. an [r, r]−graph) has a [k, k+ 1]−factor for every k satisfying 0 ≤ k ≤ r. Recently, Thomassen[3] gave a simple elegant proof to the following theorem, which is an extension of the above theorem: an [r, r + 1]−graph has a [k, k + 1]−factor for every k satisfying 0 ≤ k ≤ r. In this paper we shall first prove the next result which is an extension of the theorems mentioned above. Theorem 1. Let Gbe a graph and θ be a real number such that 0 ≤ θ ≤ 1. Suppose two integer valued functions g and f defined on V (G) satissfy g(x) < f(x) and g(x) ≤ θdG(x) ≤ f(x) (1) for all x ∈ V (G).Then G has a (g, f)−factor. Note that when g(x) ≤ f(x) instead of g(x) < f(x), the sufficient condition, which is similar to Theorem 1, for the existence of a (g, f)−factor is rather complicated [2]. If G is an [r, r + s]−graph and ks ≤ rt, then (1) in Theorem 1 holds for g(x) ≡ k, f(x) ≡ k + t and θ = kr , and we obtain the next corollary. Corollary 1. Let 0 ≤ k ≤ r, 0 ≤ s and 1 ≤ t, If ks ≤ rt, then an [r, r + s]−graph has a [k, k + 1]−factor. In the proof of Theorem 1, we shall use the following (g, f)−factor theorem due to Lovász, to which Tutte [5] gave a short proof by using his f−factor theorem. Lemma (Lovasz [1], [5, Theorem 7.2]). Let Gbe a graph and g and f be integer-valued functions defined on V (G) such that g(x) ≤ f(x) for all x ∈ V (G). Then G has a (g, f)−factor if and only if ∑ t∈T dG(t)− ∑ t∈T g(t) + ∑ s∈S f(s)− eG(S, T )− h(S, T ) ≥ 0 (2) for all disjoint subset S and T of V (G), where h(S, T )is the number of components C of G− (S ∪ T ) such that g(c) = f(c) for all c ∈ V (C) and eG(T, V (C)) + ∑ c∈V (C) f(c) ≡ 1(mod2). Note that the condition 0 ≤ g(x) ≤ f(x) ≤ dG(x) in [1], [5] can be replaced by g(x) ≤ f(x) as in the theorem. Furthermore, if g(x) < f(x) for every vertex x, then h(S, T ) = 0 and so the condition (2) becomes simple. Proof of Theorem 1. Let S, T ⊂ V (G), S ∩ T = ∅. Then h(S, T ) = 0 and we have ∑ t∈T dG(t)− ∑ t∈T g(t) + ∑ s∈S f(s)− eG(S, T ) ≥ (1− θ) ∑ t∈T dG(t)− θ ∑ s∈S f(s)− eG(S, T ≥ (1− θ)eG(T, S)− θeG(S, T )− eG(S, T ) = 0 Therefore, (2)holds and we conclude that G has a (g, f)-factor. We remark that the condition in corollary 1 is best possible. Remark.The complete bipartite graph Kr,r+s does not have a [k, k + 1]-factor for any k, t such that 0 ≤ k ≤ r, 1 ≤ t and ks ≥ rt. Proof.Let G denote the complete bipartite graph Kr,r+s. Set S and T be the partite sets of G such that |S| = r and |T | = r + s, and put g ≡ k and f ≡ k + t. Then ∑ t∈T dG(t)− ∑ t∈T g(t)+ ∑ s∈S f(s)−eG(S, T )−h(S, T ) = r(r+s)−k(r+s)+(k+t)r−r(r+s) = rt−ks < 0. Then G does not have a [k, k + t]-factor. We next give a sufficient conddition for a graph to have a factor which contains a given graph. Theorem 2. Let 0 ≤ l ≤ k ≤ r, l+ u ≤ k+ l ≤ r+ s, r 6= l, r+ s 6= l+ u, 0 ≤ s, 0 ≤ u and 1 ≤ t. Let G be an [r, r + s]−graph and H be an [l, l + u]−factor of G. If (l − k)s+ (k − r)u+ (r − l)t ≥ 0 (3) then G has a [k, k + 1]−factor which H as a graph. If s = u = t, then (3) follows, and so we obtain the next corollary. Corollary 2. Let 0 ≤ l ≤ k ≤ r and 1 ≤ t. Then an [r, r+ t]−graph has a [k, k + t]−factor which contains a given [l, l + t]−factor. Proof of Theorem 2. Let G be an [r, r + 1]−graph and H be an [l, l + t]−factor of G. Let G‘ = G− E(H) and g and f be functions from V (G‘) into the set of integers defined by g(x) = k − dH(x) and f(x) = k + t− dH(x) for all x ∈ V (G‘). If G‘ has a (g, f)−factor F , then F + E(H) is a dedired [k, k + t]−factor. Put θ = k − l r − l and λ = k + l − l − u r + s− l − u We shall show that G‘, g, f and θ satisfy (1) in Theorem 1. It is clear that 0 ≤ θ ≤ 1 and g(x) < f(x) for all x ∈ V (G‘). Since k = θr + (1− θ) ≤ θdG(x) + (1− θ)dH(x) and dG(x) = dG‘(x)− dH(x), we have g(x) ≤ θdG‘(x). Similarly, we have f(x) ≥ λdG‘(x) as k + t = λ(r + s) + (1− λ)(l + u) ≥ λdG(x) + (1− λ)dH(x). By (3), we obtain λ ≥ θ and so f(x) ≥ θdG‘(x). Consequently, (1) holds and the theorem follows. Acknowledgement The authors would like to thank Professor Hikoe Enomoto and Professor Yoshimi Egawa for their helpful discussion and valuable suggestions.