We equate the fractional factor problem with a relaxation of the famous cardinality matching problem which is one of the core problems in operation research. It has widely applications in various fields such as network design, combinatorial polyhedron and scheduling. For example, several large data packets are sent to different destinations through some channels in a data transmission network. This work helps to partition the large data packets into small ones and then make it efficiency improved. The available allocations of data packets is equated with the problem of fractional flow, which converts to fractional factor problem in a graph generated by a network.

Specifically, there is a graph that can model the complete network and the graph needs to meet the requirements that each site corresponds to a vertex and each channel corresponds to an edge in it. In normal network, the path of data transmission is selected by the shortest path between vertices. Several contributions on data transmission in networks are presented recently. Rolim *et al*. [1] studied urban sensing problem by means of opportunistic networks to support the data transmission. Vahidi *et al*. [2] provided the high-mobility airborne hyperspectral data transmission algorithm in view of unmanned aerial vehicles approach. Miridakis *et al*. [3] determined a rather cost-effective solution for the dual-hop cognitive secondary relaying system. Lee *et al*. [4] considered streaming data transmission on a discrete memoryless channel. However, in the setting of software definition network, the data transmission depends on the network flow computation. The transmission path is selected with minimum transmission congestion in the current moment. From this perspective, the model of data transmission problem in SDN is becoming what makes the fractional factor avoid certain subgraphs.

It is only the simple graphs that are chosen in our article. Let *G* = (*V*(*G*), *E*(*G*)) be a graph, in which *V*(*G*) and *E*(*G*) are the vertex set and the edge set, respectively. Let *n* = |*V*(*G*)| be the order of graph *G*. The standard notations and terminologies used but undefined clearly in this article can be found in Bondy and Mutry [5], Basavanagoud *et al*. [6], Wang [7], and Gao and Wang [8]. The *toughness t*(*G*) of a graph *G* can be stated in the below:
$$t(G)=min\{\frac{|S|}{\omega (G-S)}|S\subseteq V(G),\omega (G-S)\ge 2\},$$
if *G* is not complete; otherwise, *t*(*G*) = +∞. It is an important parameter, which is used to measure the vulnerability of networks.

Let *g* and *f* be two positive integer-valued functions on *V*(*G*) such that 0 ≤ *g*(*x*) ≤ *f*(*x*) for any *x* ∈ *V*(*G*). A *fractional* (*g*, *f*)-*factor* is a function *h* which assigns a number in [0,1] to each edge so that $g(x)\le {d}_{G}^{h}(x)\le f(x)$ for each *x* ∈ *V*(*G*), where ${d}_{G}^{h}(x)=\sum _{e\in E(x)}h(e)$ is called the *fractional* *degree* of *x* in *G*. A *fractional* (*a*, *b*)-*factor* (here *a* and *b* are both positive integers and *a* ≤ *b*) is a function *h* that assigned a number [0,1] to each edge of a graph *G* so that for each vertex *x* we have $a\le {d}_{G}^{h}(x)\le b$. Clearly, fractional (*a*, *b*)-factor is a particular case of fractional (*g*, *f*)-factor when *g*(*x*) = *a* and *f*(*x*) = *b* for any *x* ∈ *V*(*G*). If *a* = *b* = *k*(*k* ≥ 1 is an integer) for all *x* ∈ *V*(*G*), then a fractional (*a*, *b*)-factor is just a fractional *k*-factor.

It’s stated that *G* includes a Hamiltonian fractional (*g*, *f*)-factor if *G* has a fractional (*g*, *f*)-factor containing a Hamiltonian cycle. A graph *G* is said to be an ID- Hamiltonian graph if the remaining graph of *G* admits a Hamiltonian cycle, when we delete any independent set of *G*. We say that *G* has an ID-Hamiltonian fractional (*g*, *f*)-factor if the remaining graph of *G* includes a Hamiltonian fractional (*g*, *f*)-factor when we delete any independent set of *G*.

In SDN, the independent set *I* is corresponding to the website with high transmission congestion and a Hamiltonian cycle corresponding to several channels with high transmission congestion. Thus, we study the data transmission problem by considering the Hamiltonian fractional (*g*, *f*)-factor and ID-Hamiltonian fractional (*g*, *f*)- factor in the networks model. Several related results can refer to [9-14].

The sufficient conditions of Hamiltonian fractional (*g*, *f*)-factors and ID-Hamiltonian fractional (*g*, *f*)-factors in graphs are studied and at last we obtain three results that are stated in the below.

#### Theorem 1

Let *a*, *b* be two integers with 3 ≤ *a* ≤ *b*, and *G* be a Hamiltonian graph of order $n>\frac{(a+b-5)(a+b-3)}{a-2}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}g,f$ are taken to be two integer-valued functions that are defined on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). If
$$\delta (G)\ge \frac{(b-1)n+a+b-3}{a+b-3}$$
and
$$\delta (G)>\frac{(b-2)n+2\alpha (G)-1}{a+b-4},$$
then *G* has a Hamiltonian fractional (*g*, *f*)-factor.

#### Theorem 2

Let *a*, *b* be two integers with 3 ≤ *a* ≤ *b*, and *G* be a Hamiltonian graph of order *n* ≥ *b* + 2. *g*, *f* are taken to be two integer-valued functions that are defined on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). If *t*(*G*) ≥ *b* − $1+\frac{b-1}{a-2}$, then *G* has a Hamiltonian fractional (*g*, *f* )-factor.

#### Theorem 3

*a*, *b* are taken to be two integers with 3 ≤ *a* ≤ *b*, and *G* be an ID-Hamiltonian graph. And *g*, *f* are taken to be two integer-valued functions that are defined on *V*(*G*) such that *a* ≤ *g*(*x*) ≤ *f*(*x*) ≤ *b* for each *x* ∈ *V*(*G*). If
$$\begin{array}{rl}& \kappa (G)\ge max\left\{\frac{(b+1{)}^{2}+4(a-2)}{2},\right.\\ & \left.\frac{(a+b+1)(a+b-3)}{4(a-2)}+\frac{(a+b+1)(a+b-3)}{(b+1{)}^{2}}\right\}\end{array}$$(1)
and
$$\kappa (G)\ge \frac{(b+1{)}^{2}+4(a-2)}{4(a-2)}\alpha (G),$$
then *G* has an ID-Hamiltonian fractional (*g*, *f*)-factor.

The ways to prove our major results are on the basis of the listed lemmas below:

#### Lemma 1

(Liu and Zhang [15]) Since *G* is a graph and we have *H* = *G*[*T*], so *δ*(*H*) ≥ 1 and 1 ≤ *d*_{G}(*x*) ≤ *k* − 1 for every *x* ∈ *V*(*H*) where *T* ⊆ *V*(*G*) and *k* ≥ 2. Let *T*_{1},..., *T*_{k-1} be a partition of the vertices of *H* satisfying *d*_{G}(*x*) = *j* for each *x* ∈ *T*_{j} where some *T*_{j} are allowed to be empty on purpose. If each component of *H* has a vertex of degree at most *k* − 2 in *G*, then *H* has a maximal independent set *I* and a covering set *C* = *V*(*H*) − *I* such that
$$\sum _{j=1}^{k-1}(k-j){c}_{j}\le \sum _{j=1}^{k-1}(k-2)(k-j){i}_{j},$$
where *c*_{j} = |*C* ∩ *T*_{j} | and *i*_{j} = |*I* ∩ *T*_{j}| for every *j* = 1,..., *k* − 1.

#### Lemma 2

(Liu and Zhang [15]) Since *G* is a graph and *H* = *G*[*T*], so *d*_{G}(*x*) = *k* − 1 for every *x* ∈ *V*(*H*) and no component of *H* is isomorphic to *K*_{k} where *T* ⊆ *V*(*G*) and *k* ≥ 2. Then a maximal independent set *I* is obtained and the covering set *C* = *V*(*H*) − *I* of *H* satisfy
$$|V(H)|\le (k-\frac{1}{k+1})|I|,$$
and
$$|C|\le (k-1-\frac{1}{k+1})|I|.$$

#### Lemma 3

(Anstee [16]) Assume *f* and *g* be two integervalued functions that are defined on the vertex set of a graph *G* such that 0 ≤ *g*(*x*) ≤ *f*(*x*) for each *x* ∈ *V*(*G*). Then *G* has a fractional (*g*, *f*)-factor if and only if for every subset *S* of *V*(*G*), *g*(*T*) − *d*_{G-S}(*T*) ≤ *f*(*S*),where *T* = {*x* ∈ *V*(*G*)\*S* | *d*_{G-S}(*x*) ≤ *g*(*x*) − 1}.

Clearly, Lemma 3 is equal to the following version.

#### Lemma 4

Suppose that *f* and *g* are two integer-valued functions that are defined on the vertex set of a graph *G* such that 0 ≤ *g*(*x*) ≤ *f*(*x*) for each *x* ∈ *V*(*G*). Then *G* has a fractional (*g*, *f* )-factor if and only if
$$f(S)+{d}_{G-S}(T)-g(T)\ge 0$$
for all disjoint subsets *S*, *T* of *V*(*G*).

#### Lemma 5

(Katerinis [17]) If the graph *G* isn’t complete, then $t(G)\le \frac{\delta (G)}{2}$.

The tricks used to prove our main results are borrowed from Zhou et al. [18], [19] and [20], and more new technologies are introduced here to deal with complex problems.

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