The classic fractional factor problem is usually regarded as an extension of illustrious cardinality matching which is one of the hot topics in graph theory and operations research. Its extensive applications can be found in many domains like combinatorial polyhedron, scheduling and designing of network. For instance, we send some large data packets to several destinations through channels of the data transmission network, and the efficiency will be improved if the large data packets can be categorized into more smaller ones. The problem of feasibility allocations of data packets can be considered as the existence of fractional flow in the network, and it can be transformed to the fractional factor problem in the network graph.

In particular, the whole network can be modelled as a graph in which each vertex represents a site and each edge denotes a channel. In normal networks, the route of data transmission is picked out based on the shortest path between vertices. A few advances of data transmission in networks have been manifested in recent years. Rolim et al. [1] improved the data transmission by studying an urban sensing problem in view of opportunistic networks. Vahidi et al. [2] considered unmanned aerial vehicles and proposed the high-mobility airborne hyperspectral data transmission algorithm. Miridakis et al. [3] adopted a cost-effective solution to study the dual-hop cognitive secondary relaying system. Streaming data transmission on a discrete memoryless channel was discussed by Lee et al. [4]. But, in the context of software definition network (SDN), the path between vertices in data transmission relies on the current network flow computation. With minimum transmission congestion in the current moment, the transmission route is chosen. In this view, the framework of data transmission problem in SDN equals to the existence of the fractional factor avoiding certain subgraphs.

Throughout this paper, we only consider the simple graph which corresponds to a data transmission network. Let *G* = (*V*(*G*), *then E*(*G*)) be a graph with vertex set *V*(*G*) and edge set *E*(*G*). For any *x* ∈ *V*(*G*), we denote *d*_{G}(*x*) and *N*_{G}(*x*) by the degree and the open neighborhood of *x* in *G*, separately. Set *N*_{G}[*x*] = *N*_{G}(*x*) ∪ {*x*} as the closed neighborhood of *x* in *G*. We denote by *G*[*S*] the subgraph of *G* induced by *S*, and
*G*–*S* = *G*[*V*(*G*) ∖ *S*] for any *S* ⊆ *V*(*G*). For two vertex-disjoint subsets *S*, *T* ⊂ *V*(*G*), we set *e*_{G}(*S*, *T*) = |{*e* = *xy*|*x* ∈ *S*, *y* ∈ *T*}|. We denote the minimum degree of *G* by *δ*(*G*) = min{*d*_{G}(*x*) : *x* ∈ *V*(*G*)}. As a simple expression, we take *d*(*x*) to express *d*_{G}(*x*) for *x* ∈ *V*(*G*). More terminologies and notations used but undefined in our article can be found in book Bondy and Mutry [5].

Let *g* and *f* be integer-valued functions on *V*(*G*) satisfying 0 ≤ *g*(*x*) ≤ *f*(*x*) for any *x* ∈ *V*(*G*). A fractional (*g*, *f*)-factor can be considered as a function *h* that gives to every edge of a graph *G* a real number in [0, 1] with
$\begin{array}{}{\displaystyle g(x)\le \sum _{e\in E(x)}h(e)\le f(x)}\end{array}$ for each vertex *x*. If *g*(*x*) = *f*(*x*) for any *x* ∈ *V*(*G*), a fractional (*g*, *f*)-factor is a fractional *f*-factor. If *g*(*x*) = *a* and *f*(*x*) = *b* for any *x* ∈ *V*(*G*), then a fractional (*g*, *f*)-factor is a fractional [*a*, *b*]-factor. Furthermore, a fractional (*g*, *f*)-factor becomes a fractional *k*-factor if *g*(*x*) = *f*(*x*) = *k* (where *k* ≥ 1 is an integer) for any *x* ∈ *V*(*G*). Then, we always assume that *n* is the order of *G*, namely, *n* = |*V*(*G*)|.

If *G* has a fractional *p*-factor for every *p* : *V*(*G*) → ℕ such that *g*(*x*) ≤ *p*(*x*) ≤ *f*(*x*) for any *x* ∈ *V*(*G*), *G* possesses all fractional (*g*, *f*)-factors. If *g*(*x*) = *a*, *f*(*x*) = *b* for each *x* ∈ *V*(*G*) and *G* possesses all fractional (*g*, *f*)-factors, then *G* has all fractional [*a*, *b*]-factors.

Lu [6] determined the sufficient and necessary condition for a graph that has all fractional (*g*, *f*)-factors.

#### Theorem 1

(Lu [6]) Assume *G* to be a graph, *g*, *f* : *V*(*G*) → ℤ^{+} be integer functions so that *g*(*x*) ≤ *f*(*x*) for all *x* ∈ *V*(*G*), so *G* has all fractional (*g*, *f*)-factors. If and only if for any subset *S* ⊆ *V*(*G*), we have

$$\begin{array}{}{\displaystyle g(S)-f(T)+\sum _{x\in T}{d}_{G-S}(x)\ge 0,}\end{array}$$

where *T* = {*x* ∈ *V*(*G*) – *S*|*d*_{G–S}(*x*) < *f*(*x*)}.

Obviously, this sufficient and necessary condition equals to the following version.

#### Theorem 2

Assume *G* be a graph, *g*, *f* : *V*(*G*)→ ℤ^{+} be integer functions so that *g*(*x*) ≤ *f*(*x*) for all *x* ∈ *V*(*G*). As a result, *G* has all fractional (*g*, *f*)-factors. If and only if

$$\begin{array}{}{\displaystyle g(S)-f(T)+\sum _{x\in T}{d}_{G-S}(x)\ge 0}\end{array}$$

for arbitrary non-disjoint subsets *S*, *T* ⊆ *V*(*G*).

Set *g*(*x*) = *a*, *f*(*x*) = *b* for each *x* ∈ *V*(*G*), we immediately get the below corollary from Theorem 1.

#### Corollary 1

Let *G* be a graph and *a* ≤ *b* be two positive integers. Then *G* has all fractional [*a*, *b*]-factors if and only if for any subset *S* ⊆ *V*(*G*), we have

$$\begin{array}{}{\displaystyle a|S|-b|T|+\sum _{x\in T}{d}_{G-S}(x)\ge 0,}\end{array}$$

where *T* = {*x* ∈ *V*(*G*) – *S*|*d*_{G–S}(*x*) < *b*}.

Also, this sufficient and necessary condition in Corollary 1 for all fractional [*a*, *b*]-factors equals to the following version.

#### Corollary 2

Let *G* be a graph and *a* ≤ *b* be two positive integers. Then *G* has all fractional [*a*, *b*]-factors if and only if

$$\begin{array}{}{\displaystyle a|S|-b|T|+\sum _{x\in T}{d}_{G-S}(x)\ge 0}\end{array}$$

for arbitrary non-disjoint subsets *S*, *T* ⊆ *V*(*G*).

Zhou and Zhang [7] presented the sufficient and necessary situation for a graph with all fractional (*g*, *f*)-factors excluding a subgraph *H*.

#### Theorem 3

(Zhou and Zhang [7]) Let *G* be a graph and *H* be a subgraph of *G*. Let *g*, *f* : *V*(*G*) → ℤ^{+} be two integer-valued functions with *g*(*x*) ≤ *f*(*x*) for each *x* ∈ *V*(*G*). Then *G* admits all fractional (*g*, *f*)-factors excluding *H* if and only if

$$\begin{array}{}{\displaystyle g(S)-f(T)+\sum _{x\in T}{d}_{G-S}(x)\ge \sum _{x\in T}{d}_{H}(x)-{e}_{H}(S,T)}\end{array}$$

for arbitrary subset *S* of *V*(*G*), where *T* = {*x* ∈ *V*(*G*) – *S*|*d*_{G–S}(*x*) – *d*_{H}(*x*) + *e*_{H}(*x*, *S*) < *f*(*x*)}.

Clearly, this sufficient and necessary condition has the following equal version.

#### Theorem 4

Let *G* be a graph and *g*, *f* : *V*(*G*) → ℤ^{+} be two integer-valued functions with *g*(*x*) ≤ *f*(*x*) for each *x* ∈ *V*(*G*). Let *H* be a subgraph of *G*. Then *G* admits all fractional (*g*, *f*)-factors excluding *H* if and only if

$$\begin{array}{}{\displaystyle g(S)-f(T)+\sum _{x\in T}{d}_{G-S}(x)\ge \sum _{x\in T}{d}_{H}(x)-{e}_{H}(S,T)}\end{array}$$

for arbitrary non-disjoint subsets *S*, *T* ⊆ *V*(*G*).

Set *g*(*x*) = *a*, *f*(*x*) = *b* for each *x* ∈ *V*(*G*), we immediate get the Corollary below.

#### Corollary 3

Let *G* be a graph, and *a* and *b* be two integers where 1 ≤ *a* ≤ *b*. Let *H* be a subgraph of *G*. Then *G* admits all fractional [*a*, *b*]-factors excluding *H* if and only if

$$\begin{array}{}{\displaystyle a|S|+\sum _{x\in T}({d}_{G-S}(x)-{d}_{H}(x)+{e}_{H}(x,S)-b)\ge 0,}\end{array}$$

where *T* = {*x* ∈ *V*(*G*) – *S*|*d*_{G–S}(*x*) – *d*_{H}(*x*) + *e*_{H}(*x*, *S*) < *b*}.

Again, this sufficient and necessary condition can be stated as follows.

#### Corollary 4

Let *a* and *b* be two integers with 1 ≤ *a* ≤ *b*, and let *G* be a graph. Let *H* be a subgraph of *G*. So *G* admits all fractional [*a*, *b*]-factors excluding *H* if and only if

$$\begin{array}{}{\displaystyle a|S|+\sum _{x\in T}({d}_{G-S}(x)-{d}_{H}(x)+{e}_{H}(x,S)-b)\ge 0}\end{array}$$

for arbitrary non-disjoint subsets *S*, *T* ⊆ *V*(*G*).

Zhou and Sun [8] introduced the concept of all fractional (*a*, *b*, *n*′)-critical graphs. A graph *G* is named as an all fractional (*a*, *b*, *n*′)-critical graph if the remaining graph of *G* has all fractional [*a*, *b*]-factors, even though any *n*′ vertices of *G* are deleted. Also, they presented the necessary and valuable situation for a graph to be all fractional (*a*, *b*, *n*′)-critical.

#### Theorem 5

(Zhou and Sun [8]) Let *a*, *b* and *n*′ be nonnegative integers with 1 ≤ *a* ≤ *b*, and let *G* be a graph of order *n* with *n* ≥ *a* + *n*′ + 1. Then *G* is all fractional (*a*, *b*, *n*′)-critical if and only if for any *S* ⊆ *V*(*G*) with |*S*| ≥ *n*′,

$$\begin{array}{}{\displaystyle a|S|+\sum _{x\in T}{d}_{G-S}(x)-b|T|\ge a{n}^{\prime},}\end{array}$$

where *T* = {*x* : *x* ∈ *V*(*G*) ∖ *S*, *d*_{G–S}(*x*) < *b*}.

Equally, the above necessary and sufficient condition can be re-written as follows.

#### Theorem 6

Assume *a*, *b* and *n*′ be nonnegative integers with 1 ≤ *a* ≤ *b*, hence suppose *G* is a graph of order *n* with *n* ≥ *a* + *n*′ + 1. Then, *G* is all fractional (*a*, *b*, *n*′)-critical if and only if

$$\begin{array}{}{\displaystyle a|S|+\sum _{x\in T}{d}_{G-S}(x)-b|T|\ge a{n}^{\prime}}\end{array}$$

for arbitrary non-disjoint subsets *S*, *T* ⊆ *V*(*G*) with |*S*| ≥ *n*′.

More results from this topic, regarding fractional factor, fractional deleted graphs, fractional critical and other network applications can be found in Zhou et al. [9], Jin [10], Gao and Wang [11] and [12], Gao et al. [13], [14] and [15], and Guirao and Luo [16].

In this paper, we first introduce some new concepts. A graph *G* is named as an all fractional (*g*, *f*, *m*)-deleted graph if the remaining graph of *G* has all fractional (*g*, *f*)-factors, when any *m* edges of *G* are deleted. If for any *x* ∈ *V*(*G*), we have *g*(*x*) = *a* and *f*(*x*) = *b*, then an all fractional (*g*, *f*, *m*)-deleted graph becomes an all fractional (*a*, *b*, *m*)-deleted graph, i.e., a graph *G* is named as an all fractional (*a*, *b*, *m*)-deleted graph if the remaining graph of *G* has all fractional [*a*, *b*]-factors, when any *m* edges of *G* are deleted. Following this definition, we clearly see that Theorem 3 and Theorem 4 show the necessary and sufficient condition of all fractional (*g*, *f*, *m*)-deleted graph with |*E*(*H*)| = *m* respectively, and Corollary 3 and Corollary 4 present the necessary and sufficient condition of all fractional (*a*, *b*, *m*)-deleted graph respectively.

Next, we combine two concepts, all fractional (*g*, *f*, *m*)-deleted graph and all fractional (*g*, *f*, *m*)-critical graph together. A graph *G* is named as an all fractional (*g*, *f*, *n*′, *m*)-critical deleted graph if the remaining graph of *G* is still an all fractional (*g*, *f*, *m*)-deleted graph, when any *n*′ vertices of *G* are deleted. If *g*(*x*) = *a*, *f*(*x*) = *b* for each *x* ∈ *V*(*G*), every fractional (*g*, *f*, *n*′, *m*)-critical deleted graph becomes all fractional (*a*, *b*, *n*′, *m*)-critical deleted graph, it means, after deleting any *n*′ vertices of *G* the remaining graph of *G* is still an all fractional (*a*, *b*, *m*)-deleted graph.

The concept of all fractional (*g*, *f*, *n*′, *m*)-critical deleted graph reflects the feasibility of data transmission in data transmission networks, in the case of some sites or channels being damaged. At the same time, it also provides theoretical support for SDN: in the peak data transmission, some sites and communications are often in the situation of information congestion, so we regard the site and channel in a blocked state as the vertices and edges that need to be deleted. Then, we consider the existence of fractional factors in the resulting subgraph. Thus, it inspires us to study the sufficient condition of all fractional (*g*, *f*, *n*′, *m*)-critical deleted graph from the graph structure prospect. It will imply which structures of network can ensure the success of data transmission, and the theoretical results obtained in our article can help us carry out the network design.

In this paper, we explore relations between the neighborhood union condition and all fractional (*g*, *f*, *n*′, *m*)-critical deleted graphs. The first key result can be formulated as follows.

#### Theorem 7

Assume *a*, *b*, *n*′ and *m* are four non-negative integers satisfying 2 ≤ *a* ≤ *b*. Assume *G* to be a graph with
$\begin{array}{}{\displaystyle \delta (G)\ge \frac{(a+b-1{)}^{2}+4b}{4a}+m+{n}^{\prime}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}n\ge \frac{2(a+b)(a+b+m-1)}{a}+{n}^{\prime}.}\end{array}$ Let *g* and *f* be two integer functions which are defined on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). If |*N*_{G}(*u*) ∪ *N*_{G}(*v*)| ≥
$\begin{array}{}{\displaystyle \frac{bn+a{n}^{\prime}}{a+b}}\end{array}$ for arbitrary two nonadjacent vertices *u* and *v* in *G*, then *G* is all fractional (*g*, *f*, *n*′, *m*)-critical deleted.

Set *n*′ = 0 in Theorem 7, then we get the following corollary.

#### Corollary 5

Assume *a*, *b* and *m* be three non-negative integers satisfying 2 ≤ *a* ≤ *b*. Assume *G* is a graph with
$\begin{array}{}{\displaystyle \delta (G)\ge \frac{(a+b-1{)}^{2}+4b}{4a}+m}\end{array}$ and
$\begin{array}{}{\displaystyle n\ge \frac{2(a+b)(a+b+m-1)}{a}.}\end{array}$ Let *g* and *f* be two integer functions whose definition can be found on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). If |*N*_{G}(*u*) ∪ *N*_{G}(*v*)| ≥
$\begin{array}{}{\displaystyle \frac{bn}{a+b}}\end{array}$ for arbitrary two nonadjacent vertices *u* and *v* in *G*, then *G* is all fractional (*g*, *f*, *m*)-deleted.

By setting *m* = 0 in Theorem 7, the following corollary is obtained.

#### Corollary 6

Assume *a*, *b* and *n*′ to be three non-negative integers satisfying 2 ≤ *a* ≤ *b*. Let *G* be a graph with *δ*(*G*) ≥
$\begin{array}{}{\displaystyle \frac{((a+b-1{)}^{2}+4b)}{4a}+{n}^{\prime}}\end{array}$ and
$\begin{array}{}{\displaystyle n\ge \frac{2(a+b)(a+b-1)}{a}+{n}^{\prime}.}\end{array}$ Let *g* and *f* be two integer functions whose definition can be found on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). If |*N*_{G}(*u*) ∪ *N*_{G}(*v*)| ≥
$\begin{array}{}{\displaystyle \frac{bn+a{n}^{\prime}}{a+b}}\end{array}$ for arbitrary two nonadjacent vertices *u* and *v* in *G*, then *G* is all fractional (*g*, *f*, *n*′)-critical.

Set *g*(*x*) = *a* and *f*(*x*) = *b* for any *x* ∈ *V*(*G*), then we get the following condition for all fractional (*a*, *b*, *n*′, *m*)-critical deleted graph.

#### Corollary 7

Let *a*, *b*, *n*′ and *m* be four non-negative integers satisfying 2 ≤ *a* ≤ *b*. Let *G* be a graph with
$\begin{array}{}{\displaystyle \delta (G)\ge \frac{(a+b-1{)}^{2}+4b}{4a}}\end{array}$ + *m* + *n*′ and
$\begin{array}{}{\displaystyle n\ge \frac{2(a+b)(a+b+m-1)}{a}+{n}^{\prime}.}\end{array}$ If |*N*_{G}(*u*) ∪ *N*_{G}(*v*)| ≥
$\begin{array}{}{\displaystyle \frac{bn+a{n}^{\prime}}{a+b}}\end{array}$ for arbitrary two nonadjacent vertices *u* and *v* in *G*, then *G* is all fractional (*a*, *b*, *n*′, *m*)-critical deleted.

Set *n*′ = 0 in Corollary 7, then the corollary below can be obtained.

#### Corollary 8

Let *a*, *b* and *m* be three non-negative integers satisfy 2 ≤ *a* ≤ *b*. Let *G* be a graph with
$\begin{array}{}{\displaystyle \delta (G)\ge \frac{(a+b-1{)}^{2}+4b}{4a}}\end{array}$ + *m* and *n* ≥
$\begin{array}{}{\displaystyle \frac{2(a+b)(a+b+m-1)}{a}}\end{array}$. If |*N*_{G}(*u*) ∪ *N*_{G}(*v*)| ≥
$\begin{array}{}{\displaystyle \frac{bn}{a+b}}\end{array}$ for arbitrary two nonadjacent vertices *u* and *v* in *G*, then *G* is all fractional (*a*, *b*, *m*)-deleted.

Set *m* = 0 in Corollary 7, then the following corollary is obtained.

#### Corollary 9

Let *a*, *b* and *n*′ be three non-negative integers satisfy 2 ≤ *a* ≤ *b*. Let *G* be a graph with *δ*(*G*) ≥
$\begin{array}{}{\displaystyle \frac{(a+b-1{)}^{2}+4b}{4a}}\end{array}$ + *n*′ and
$\begin{array}{}{\displaystyle n\ge \frac{2(a+b)(a+b-1)}{a}+{n}^{\prime}.}\end{array}$ If |*N*_{G}(*u*) ∪ *N*_{G}(*v*)| ≥
$\begin{array}{}{\displaystyle \frac{bn+a{n}^{\prime}}{a+b}}\end{array}$ for arbitrary two nonadjacent vertices *u* and *v* in *G*, then *G* is all fractional (*a*, *b*, *n*′)-critical.

Our second main result is stated as follows.

#### Theorem 8

Let *a*, *b*, *n*′ and *m* be four non-negative integers satisfying 2 ≤ *a* ≤ *b*. Let *G* be a graph with order
$\begin{array}{}{\displaystyle n\ge \frac{(a+2b-3)(a+b-2)}{a}+\frac{a{n}^{\prime}}{a-1}+m.}\end{array}$ Let *g* and *f* be two integer functions whose definition can be found on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). Suppose that

$$\begin{array}{}{\displaystyle {N}_{G}(X)=V(G)\phantom{\rule{1em}{0ex}}\mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|\ge \lfloor \frac{(an-a-a{n}^{\prime}-2m)n}{(a+b-1)(n-1)}\rfloor ;}\end{array}$$

or

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{N}_{G}(X)\ge \frac{(a+b-1)(n-1)}{an-a-a{n}^{\prime}-2m}|X|}\\ {\displaystyle \mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|<\lfloor \frac{(an-a-a{n}^{\prime}-2m)n}{(a+b-1)(n-1)}\rfloor}\end{array}$$

for any subset *X* ⊆ *V*(*G*). Then *G* is all fractional (*g*, *f*, *n*′, *m*)-critical deleted.

Set *n*′ = 0 in Theorem 8, then we yield the following corollary.

#### Corollary 10

Let *a*, *b*, and *m* be three non-negative integers satisfy 2≤ *a* ≤ *b*. Let *G* be a graph with order *n* ≥
$\begin{array}{}{\displaystyle \frac{(a+2b-3)(a+b-2)}{a}}\end{array}$ + *m*. Let *g* and *f* be two integer functions defined on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). Suppose that

$$\begin{array}{}{\displaystyle {N}_{G}(X)=V(G)\phantom{\rule{1em}{0ex}}\mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|\ge \lfloor \frac{(an-a-2m)n}{(a+b-1)(n-1)}\rfloor ;}\end{array}$$

or

$$\begin{array}{}{\displaystyle {N}_{G}(X)\ge \frac{(a+b-1)(n-1)}{an-a-2m}|X|}\\ {\displaystyle \mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|<\lfloor \frac{(an-a-2m)n}{(a+b-1)(n-1)}\rfloor}\end{array}$$

for any subset *X* ⊆ *V*(*G*). Then *G* is all fractional (*g*, *f*, *m*)-deleted.

Set *m* = 0 in Theorem 8, then the Corollary below can be inferred.

#### Corollary 11

Let *a*, *b* and *n*′ be three non-negative integers satisfy 2≤ *a* ≤ *b*. Let *G* be a graph with order *n* ≥
$\begin{array}{}{\displaystyle \frac{(a+2b-3)(a+b-2)}{a}+\frac{a{n}^{\prime}}{a-1}.}\end{array}$ Let *g* and *f* be two integer functions defined on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). Suppose that

$$\begin{array}{}{\displaystyle {N}_{G}(X)=V(G)\phantom{\rule{1em}{0ex}}\mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|\ge \lfloor \frac{(an-a-a{n}^{\prime})n}{(a+b-1)(n-1)}\rfloor ;}\end{array}$$

or

$$\begin{array}{}{\displaystyle {N}_{G}(X)\ge \frac{(a+b-1)(n-1)}{an-a-a{n}^{\prime}}|X|}\\ {\displaystyle \mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|<\lfloor \frac{(an-a-a{n}^{\prime})n}{(a+b-1)(n-1)}\rfloor}\end{array}$$

for any subset *X* ⊆ *V*(*G*). Then *G* is all fractional (*g*, *f*, *n*′)-critical.

Set *g*(*x*) = *a* and *f*(*x*) = *b* for any *x* ∈ *V*(*G*), then we get the following condition for an all fractional (*a*, *b*, *n*′, *m*)-critical deleted graph.

#### Corollary 12

Let *a*, *b*, *n*′ and *m* be four non-negative integers satisfying 2 ≤ *a* ≤ *b*. Let *G* be a graph with order
$\begin{array}{}{\displaystyle n\ge \frac{(a+2b-3)(a+b-2)}{a}+\frac{a{n}^{\prime}}{a-1}+m.}\end{array}$ Suppose that

$$\begin{array}{}{\displaystyle {N}_{G}(X)=V(G)\phantom{\rule{1em}{0ex}}\mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|\ge \lfloor \frac{(an-a-a{n}^{\prime}-2m)n}{(a+b-1)(n-1)}\rfloor ;}\end{array}$$

or

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}{N}_{G}(X)\ge \frac{(a+b-1)(n-1)}{an-a-a{n}^{\prime}-2m}|X|}\\ {\displaystyle \mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|<\lfloor \frac{(an-a-a{n}^{\prime}-2m)n}{(a+b-1)(n-1)}\rfloor}\end{array}$$

for any subset *X* ⊆ *V*(*G*). Then *G* is all fractional (*a*, *b*, *n*′, *m*)-critical deleted.

Set *n*′ = 0 in Corollary 12, then we have the following corollary.

#### Corollary 13

Let *a*, *b* and *m* be three non-negative integers satisfy 2 ≤ *a* ≤ *b*. Let *G* be a graph with order *n* ≥
$\begin{array}{}{\displaystyle \frac{(a+2b-3)(a+b-2)}{a}}\end{array}$ + *m*. Suppose that

$$\begin{array}{}{\displaystyle {N}_{G}(X)=V(G)\phantom{\rule{1em}{0ex}}\mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|\ge \lfloor \frac{(an-a-2m)n}{(a+b-1)(n-1)}\rfloor ;}\end{array}$$

or

$$\begin{array}{}{\displaystyle {N}_{G}(X)\ge \frac{(a+b-1)(n-1)}{an-a-2m}|X|}\\ \mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|<\lfloor \frac{(an-a-2m)n}{(a+b-1)(n-1)}\rfloor \end{array}$$

for any subset *X* ⊆ *V*(*G*). Then *G* is all fractional (*a*, *b*, *m*)-deleted.

Set *m* = 0 in Corollary 12, then we deduce the following corollary.

#### Corollary 14

Let *a*, *b* and *n*′ be three non-negative integers satisfying 2 ≤ *a* ≤ *b*. Let *G* be a graph with order *n* ≥
$\begin{array}{}{\displaystyle \frac{(a+2b-3)(a+b-2)}{a}+\frac{a{n}^{\prime}}{a-1}.}\end{array}$ Suppose that

$$\begin{array}{}{\displaystyle {N}_{G}(X)=V(G)\phantom{\rule{1em}{0ex}}\mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|\ge \lfloor \frac{(an-a-a{n}^{\prime})n}{(a+b-1)(n-1)}\rfloor ;}\end{array}$$

or

$$\begin{array}{}{\displaystyle {N}_{G}(X)\ge \frac{(a+b-1)(n-1)}{an-a-a{n}^{\prime}}|X|}\\ {\displaystyle \mathrm{i}\mathrm{f}\phantom{\rule{1em}{0ex}}|X|<\lfloor \frac{(an-a-a{n}^{\prime})n}{(a+b-1)(n-1)}\rfloor}\end{array}$$

for any subset *X* ⊆ *V*(*G*). Then *G* is all fractional (*a*, *b*, *n*′)-critical deleted.

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