In an NFV network, the availability of resource scheduling can be transformed to the existence of the fractional factor in the corresponding NFV network graph. Researching on the existence of special fractional factors in network structure can help to construct the NFV network with efficient application of resources. Let h: E(G) → [0, 1] be a function. We write \({d}_{G}^{h}(x)=\sum \limits_{e\ni x}h(e)\). We call a graph F_h with vertex set V(G) and edge set E_h a fractional (g, f)-factor of G with indicator function h if \(g(x)\le {d}_{G}^{h}(x)\le f(x)\) holds for any x ∈ V(G), where E_h = {e : e ∈ E(G), h(e) > 0}. We say that G has property E(m, n) with respect to a fractional (g, f)-factor if for any two sets of independent edges M and N with ∣M∣ = m, ∣N∣ = n, and \(M\cap N=\varnothing \), G admits a fractional (g, f)-factor F_h with h(e) = 1 for any e ∈ M and h(e) = 0 for any e ∈ N. The concept of property E(m, n) with respect to a fractional (g, f)-factor corresponds to the structure of an NFV network where certain channels are occupied or damaged in some period of time. In this paper, we consider the resource scheduling problem in NFV networks using graph theory, and show a neighborhood union condition for a graph to have property E(1, n) with respect to a fractional (g, f)-factor. Furthermore, it is shown that the lower bound on the neighborhood union condition in the main result is the best possible in some sense.

在NFV网络中，可以将资源调度的可用性转换为相应NFV网络图中分数因子的存在。对网络结构中特殊分数因子的存在进行研究有助于构建有效利用资源的NFV网络。令h：  E（G）→[0，1]为一个函数。我们写\（{d} _ {G} ^ {h}（x）= \ sum \ limits_ {e \ ni x} h（e）\）。我们称顶点集为V（G）的图F _h和边缘集为E _h的具有指标函数的G的分数（g，  f）因子ħ如果\（G（X）\文件{d} _ {G} ^ {H}（X）\文件F（X）\）适用于任何X  ∈  V（G ^），其中È _ħ  = { Ë  ：È  ∈  è（ģ），  ħ（ê）> 0}。我们说，如果对于任意两套独立边M和N，且∣ M ∣ =  m，N N ∣ =  ，则G关于分数（g，  f）因子具有属性E（m，  n）。Ñ，和\（M \帽N = \ varnothing \） ，G ^承认分数（克，  ˚F） -因子˚F _ħ与ħ（ë）对于任何= 1 Ë  ∈ 中号和ħ（ë）对于任何= 0 ë  ∈  ñ。关于分数（g，  f）的性质E（m，  n）的概念因子对应于NFV网络的结构，其中某些频道在一段时间内被占用或损坏。在本文中，我们使用图论考虑了NFV网络中的资源调度问题，并给出了图相对于分数（g，  f）因子具有属性E（1，  n）的邻域联合条件。此外，表明在某种意义上，主要​​结果中邻域联合条件的下限是最佳的。