Sufficient Conditions for a Graph to Have All [a, b]-Factors and (a, b)-Parity Factors

Let G be a graph with vertex set V and let b>a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>a$$\end{document} be two positive integers. We say that G has all [a, b]-factors if G has an h-factor for every h:V→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h: V\rightarrow {\mathbb {N}}$$\end{document} such that a≤h(v)≤b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \le h(v) \le b$$\end{document} for every v∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in V$$\end{document} and ∑v∈Vh(v)≡0(mod2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{v\in V}h(v)\equiv 0\pmod 2$$\end{document}. A spanning subgraph F of G is called an (a, b)-parity factor, if dF(v)≡a≡b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_F (v) \equiv a \equiv b $$\end{document} (mod 2) and a≤dF(v)≤b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \le d_F (v) \le b$$\end{document} for all v∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in V$$\end{document}. In this paper, we have developed sufficient conditions for the existence of all [a, b]-factors and (a, b)-parity factors of G in terms of the independence number and connectivity of G. This work extended an earlier result of Nishimura (J Graph Theory 13: 63–69, 1989). Furthermore, we show that these results are best possible in some cases.


Introduction
Many real-world networks can be conveniently modeled by graphs and studied by graph theory methods. A network can be defined as a graph in which vertices and B Yuqing Lin yuqing.lin@newcastle.edu.au 1 edges of the graph correspond to nodes and links between the nodes in the network. Graph connectivity is an important parameter of graphs and networks. Through the study of the graph connectivity, people have a further understanding of the structure and properties of graphs, which then results in solutions to many practical problems such as reliable communication network design. Independence number and factors in graphs have attracted a great deal of attention due to their applications in matching theory, network design, combinatorial design, etc. In this paper, we study the existence of degree constraint factors in terms of independence number and connectivity.
All graphs considered in this paper are simple. Let G be a graph with vertex set V (G), edge set E(G) and order n = |V (G)|. Given a vertex v ∈ V (G), let N G (v) denote the set of vertices adjacent to v in G and d G For any subset X ⊆ V (G), let G[X ] denote the vertex induced subgraph of G induced by X and the subgraph induced by vertex set V (G) − X is also denoted by G − X . For vertex v ∈ V (G), the number of edges which link v and X is denoted The cardinality of the smallest cut-set in G is called the connectivity of G and is denoted by κ(G). A subset S of V (G) is called an independent set of graph G if any two vertices of S are non-adjacent in G. A maximum independent set is an independent vertex set of a graph G containing the largest possible number of vertices; the cardinality of this set is called the independence number of graph G and denoted by α(G). The join G = G 1 + G 2 is the graph obtained from two vertex disjoint graphs G 1 and G 2 , by joining each vertex in G 1 to every vertex in G 2 .
Let g and f be two integer-valued functions defined on V (G) such that 0 ≤ g(x) ≤ f (x) for all x ∈ V (G). We will say that G has all (g, f )-factors if G has an h-factor for every h : , then an f -factor is a k-factor. A (g, f )-parity factor is called an (a, b)-parity factor if g(x) ≡ a and f (x) ≡ b. For any real function h on V (G) and any subset S ⊆ V (G), we denote The following theorem gives a criteria for a graph G to have a (g, f )-parity factor. Theorem 1.1 (Lovász,[6]) A graph G has a (g, f )-parity factor if and only if for any disjoint subsets S, T of V (G), where τ (S, T ) denotes the number of connected components C of G − (S ∪ T ) such that e G (V (C), T ) + g(V (C)) ≡ 1 (mod 2).
There are some sufficient conditions for a graph to have all [a, b]-factors [2,3]. In particular, Niessen [8] gave a characterization for a graph to have all (g, f )-factors.

Theorem 1.2 (Niessen, [8]) A graph G has all (g, f )-factors if and only if
Lu et al. [7] showed that it is NP-hard to determine whether a graph has all (g, f )factors, which answers a problem proposed by Niessen [8].
A classical result of Chvátal and Erdős [1] says that every graph G whose vertex connectivity is at least as large as its independence number is Hamiltonian. Motivated by Chvátal and Erdős's result, Nishimura [9] obtained some sufficient conditions for a graph to have a k-factor.

Theorem 1.4 (Nishimura, [9]) Let k ≥ 2 be an even integer and G be a graph. If
For the existence of f -factors, a sufficient condition on the independence number and connectivity was given by Katerinis and Tsikopoulos [4]. Kouider and Lonc [5] gave a sufficient condition for the existence of [a, b]-factors in terms of the minimum degree, the connectivity and the independence number. Furthermore, Zhou [11] obtained a sufficient condition for the existence of [a, b]-factors based on independence number and connectivity.
In this paper, we generalize Nishimura's results in two aspects. Firstly, we extend extended the results from k-factors to all [a, b]-factors and obtained sufficient conditions for a graph to have all [a, b]-factors in terms of the independence number and the connectivity. The result is stated in the following.

Theorem 1.5 Let G be a graph, and let b > a ≥ 1 be two integers. If one of the following two conditions holds, then G contains all [a, b]-factors. (i) b is odd and
(ii) b is even and Secondly, we have to extend Nishimura's results to (a, b)-parity factors, and the result is listed below.

Theorem 1.6 Let b, a be two positive integers such that b > a and b ≡ a (mod 2).
Let G be a graph such that b|V (G)| is even. If one of the following two conditions holds, then G contains an (a, b)-parity factor.
(i) b is odd and " in Theorem 1.5 (i) is best possible in some cases. Let K p denote the complete graph with p vertices. Let a, b, s, k be By Theorem 1.2, the property that G contains all (a, b)-factors does not hold. Let a, b, s, k be four positive integers such that − 1/a is not an integer. Consider the join G = K s + k K b+2 2 . Along the same line of reasoning, we can see that the condition " in Theorem 1.5 (ii) is also best possible in some cases.

Proofs of Theorem 1.5
Proof of Theorem 1.5. Firstly, suppose that condition (i) holds and the theorem does not hold. Since b > a, by Theorem 1.2, there exist two disjoint vertex subsets S and where s = |S|, t = |T | and q := q(S, T ) denotes the number of connected components of G − (S ∪ T ). If S ∪ T = ∅, we have q ≥ 2 by (1), which implies that G consists of at least two connected components, which contradicts to the condition that κ(G) ≥ 1. So we assume that S ∪T = ∅. If q ≥ 1, let C 1 , C 2 , . . . , C q denote the connected components of G −(S ∪T ). We choose S and T satisfying (1) and, subject to this, S ∪T is maximal. For convenience, let W := 1≤i≤q V (C i ) and p := κ(G − S). Now we claim that T = ∅. Suppose that T = ∅. By (1) and the definition of W , we have α(G) ≥ q ≥ as + 2. Note that κ(G) ≤ s and (b + 1) 2 > 4. So we have We continue these procedures until we reach the situation where V (T j+1 ) = ∅ for some j. Without loss of generality, suppose that N k = ∅ and N k+1 = ∅, where k ≥ 1. Then we have a sequence of non-empty sets of vertices N 1 , . . . , N k . Let n j := |N j | for j ∈ {1, 2, ..., k}. From the discussion, we can infer that and since all vertices in N j have degree at least n j − 1 in T j , thus Note that an edge joining a vertex x in N j and a vertex y in N l (1 ≤ j < l ≤ k) is counted only once; that is to say, it is counted in the term d T j (x) but not in the term d T l (y). From this observation and (3), we have On the other hand, since G − S is p-connected, then for each Summing up these inequalities for all j, we have Combining (4) and (5), we get Thus we have, i.e., Suppose that there exists a vertex u ∈ W such that one of following two conditions holds: contradicting to the maximality of S ∪ T . This completes the proof of Claim 2.
Since b > a, by Claims 1 and 2, G[W ] does not contain isolated vertices. Hence Note that q = r 1 + r 2 , and κ(G) ≤ s + p.
To show the theorem is true when condition (ii) holds, since n j ( (7), then following the same discussion as for Theorem 1.5 (i), we can obtain the proof of Theorem 1.5 (ii).

Proof of Theorem 1.6
Proof of Theorem 1.6. The proof of this theorem is similar to the proof of Theorem 1.5, so we omit some details in the proof just to avoid repeating ourselves. Firstly, suppose that condition (i) holds and G contains no (a, b)-parity factors. By Theorem 1.1, there exist two disjoint subsets S and T of V (G) such that where s := |S|, t := |T | and τ := τ (S, T ) denotes the number of the connected With the same discussion of the proof of Theorem 1.5, we may assume that S ∪ T = ∅. We choose S and T satisfying (11) and S ∪ T is maximal. Now we claim that T = ∅. Suppose that T = ∅. By (11) and the definition of W , we have α(G) ≥ τ ≥ bs + 2. Note that κ(G) ≤ s and (a + 1) 2 ≥ 4. So we have a contradiction. Let p, x 1 , . . . , x k , n 1 , . . . , n k be defined as in the proof of Theorem 1.5. With the same discussion as (6) in Theorem 1.5, we also have Next, we estimate e G (T , W ).

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