A Sufficient Condition for the Existence of Restricted Fractional (gf)-Factors in Graphs

Abstract

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 hE(G) → [0, 1] be a function. We write \({d}_{G}^{h}(x)=\sum \limits_{e\ni x}h(e)\). We call a graph Fh with vertex set V(G) and edge set Eh a fractional (gf)-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 Eh = {e :  e ∈ E(G), h(e) > 0}. We say that G has property E(mn) with respect to a fractional (gf)-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 (gf)-factor Fh with h(e) = 1 for any e ∈ M and h(e) = 0 for any e ∈ N. The concept of property E(mn) with respect to a fractional (gf)-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 (gf)-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.

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Acknowledgments

The authors would like to thank the reviewers for their constructive comments in improving the quality of this paper.

Funding

This work is supported by the Six Talent Peaks Project in Jiangsu Province, China, grant no. JY-022.

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Zhou, S., Sun, Z. & Pan, Q. A Sufficient Condition for the Existence of Restricted Fractional (gf)-Factors in Graphs. Probl Inf Transm 56, 332–344 (2020). https://doi.org/10.1134/S0032946020040043

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Keywords

  • NFV network
  • graph
  • neighborhood union
  • fractional (gf)-factor
  • restricted fractional (gf)-factors