Improved Independent Set Conditions for Fractional Factors

A graph G is called a fractional (g, f, n′,m)-critical deleted graph if after deleting any n′ vertices from G, the resulting graph admits a fractional (g, f,m)-deleted graph. A graph G is called a fractional ID-(g, f,m)-deleted if after deleting any independent set I from G, the resulting graph admits a fractional (g, f,m)-deleted graph. In this paper, we improve independent set conditions for a graph to be fractional (g, f, n′,m)-critical deleted and fractional ID-(g, f,m)-deleted. Furthermore, we present some examples to show the sharpness of given independent set bounds.


Introduction
In computer network, the fractional factor theory is used to test the feasibility of data transmission, and thus raises great attention from researchers. The purpose of this paper is to consider the data transmission problem from graph theory point of view, and only simple graphs are considered here. Let G = (V (G), E(G)) be a graph with its vertex set V (G) and its edge set E(G), and n = |V (G)| is the order of graph. Set e G (S, T ) = |{e = uv : u ∈ S, v ∈ T }| for any non-disjoint S, T ⊂ V (G). The standard notations and terminologies in this paper can be referred to Bondy and Murty [1].
Let g and f be two integer-valued functions defined on vertex set satisfying 0 ≤ g(x) ≤ f (x) for any x ∈ V (G). A fractional (g, f )-factor is a real function h(e) ≤ f (x) for each vertex x. A fractional f -factor is a special case of fractional (g, f )-factor if g(x) = f (x); a fractional [a, b]-factor if g(x) = a, f (x) = b; a fractional k-factor if g(x) = f (x) = k for arbitrarily x ∈ V (G), respectively. There are different names of graph in the different setting: fractional (g, f, m)-deleted graph (still has a fractional (g, f )-factor after deleting any m edges); fractional (g, f, n )-critical graph (still has a fractional (g, f )-factor after deleting any n vertices); fractional (g, f, n , m)-critical deleted graph (still a fractional (g, f, m)-deleted graph after removing any n vertices from G). And, the corresponding fractional [a, b]-, f -and k-graphs can be well defined. In particular, if the deleted vertex set is an independent set, then it called fractional ID-(g, f, m)-deleted graph (if G − I is a fractional (g, f, m)-deleted graph for any independent set I of G), and the corresponding fractional ID-(f, m)-deleted graph, fractional ID-(a, b, m)-deleted graph and fractional ID-(k, m)-deleted graph can be well defined. Some results on sufficient conditions for the existence of fractional factor can be referred to Gao and Wang [6] and [7], Wu et al. [8] and [9], and Zhou et al. [10], [11], [12], [13], [14] and [15].
Our main results are stated as follows which extend the previous results manifested in Gao et al. [3], [4] and [5]. The sharpness of the bounds will be presented in Section (3), and the detailed proofs will be presented in the next section. Theorem 1.1 Let G be a graph of order n. Let a, b, n , m, ∆ be five integers with i ≥ 2, 2 ≤ a ≤ b − ∆ and n , m, ∆ ≥ 0. Let g, f be two integer-valued functions for any independent subset {x 1 , x 2 , · · · , x i } of V (G), then G is a fractional (g, f, n , m)-critical deleted graph.
Theorem 1.2 Let G be a graph of order n. Let a, b, n , m, ∆ be five integers with a + ∆ + n , and for any independent subset {x 1 , x 2 , · · · , x i } of V (G), then G is a fractional (g, f, n , m)-critical deleted graph.
Set n = 0 in Theorem (1.1) and Theorem (1.2), then the above two conclusions become Theorem 1 and Theorem 2 in Gao et al. [3] which reveal the independent set conditions for fractional (g, f, m)-deleted graphs.
Corollary 1.4 Let G be a graph of order n. Let a, b, m, ∆ be five integers with a + ∆ , and for any independent subset {x 1 , x 2 , · · · , x i } of V (G), then G is a fractional (g, f, m)-deleted graph.
Taking m = 0 in Theorem (1.1) and Theorem (1.2), we obtain the corresponding independent set conditions for fractional (g, f, n )-critical graphs. Furthermore, by setting g(x) = a and f (x) = b, the corresponding independent set conditions for fractional (a, b, n , m)-critical deleted graphs are analyzed. We don't list these corollaries one by one. Moreover, we don't consider the setting in g(x) = f (x) for each x ∈ V (G) due to the corresponding conclusions on ∆ = 0 are already manifested in other published papers.
When it comes to fractional fractional ID-(g, f, m)-deleted graph setting, we determine the following two theorems on independent set degree condition and independent set neighborhood union condition, respectively.
Theorem 1.6 Let G be a graph of order n, and let a, b, i, m, ∆ be five nonnegative The corresponding sufficient conditions for fractional ID-(g, f )-factor-critical graphs can be yield by setting m = 0 in Theorem (1.5) and Theorem (1.6), respectively. In particular, in the case g(x) = a and f (x) = b for each x ∈ V (G), the independent set conditions for fractional ID-(a, b, m)-deleted graph can be derived. Other setting regard ∆ = 0 are not considered in this paper. Proof of our main results mainly depended on the following lemma which present the necessary and sufficient condition of a graph to be fractional (g, f, n , m)-critical deleted.
Lemma 1.7 (Gao [2]) Let G be a graph, g, f be two integer-valued functions defined on V (G) such that g(x) ≤ f (x) for each x ∈ V (G). Let n , m be two non-negative integers. Then G is fractional (g, f, n , m)-critical deleted graph if and only if By taking n = 0 in Lemma (1.7), we deduce the necessary and sufficient condition of a graph to be fractional (g, f, m)-deleted which will be used to explain the sharpness of Theorem (1.5) and Theorem (1.6).
for all disjoint subsets S, T of V (G).

Proof of Main Results
The purpose of this section is to present the detailed proofing procedures of four results manifested in last section. The tricks used in this section are mainly followed by Gao et al. [3], [4] and [5].
where |S| ≥ n = |U |. We select subsets S and T such that |T | is minimal. Obviously, and select This produces a contradiction.
In light of independent set neighborhood union condition described in the theorem, we infer Since By means of (2), If d i = 0, then d 1 = · · · = d i = 0. In view of (2), we get we get |S| ≥ (b − ∆)n + (a + ∆)n a + b and |T | ≤ n − |S| ≤ (a + ∆)n − (a + ∆)n a + b . By means of is also a contradiction. Therefore, the desired theorem is proved.
In light of independent set neighborhood union condition described in the theorem, we infer Using the same fashion, we have (6) By means of (5), (6), If d i = 0, then d 1 = · · · = d i = 0. In view of (5), we get |S| ≥ (b − ∆)n + (a + ∆)n a + b and |T | ≤ n − |S| ≤ (a + ∆)n − (a + ∆)n a + b . By means of is also a contradiction. Therefore, the desired theorem is proved. Here we only present the proof of Theorem (1.6), and the proof of Theorem (1.5) can be done by using the similar techniques.
Set G = G − I for any independent set I. If |I| = 1, then for any independent subset {x 1 , x 2 , · · · , x i } of G . Thus, the result holds from Corollary (1.4).
We now discuss the case |I| ≥ 2. By means of independent set neighborhood for any independent subset {x 1 , x 2 , · · · , x i } of V (G) and |I| ≥ 2. Therefore, we have In all, Theorem (1.6) is hold.

Sharpness
The aim of this section is to present that the independent set results in Theorem (1.1)-(1.6) are tight.
Again, the examples presented in Section (3) also show that the independent set degree and neighborhood union conditions are best possible.