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Some Existence Theorems on All Fractional (g, f)-factors with Prescribed Properties

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Abstract

Let G be a graph, and g, f: V (G) → Z+ with g(x) ≤ f(x) for each x ∈ V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) → Z+ with g(x) ≤ r(x) ≤ f(x) for any x ∈ V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) → Z+ with g(x) ≤ r(x) ≤ f(x) for all x ∈ V (G), G has a fractional r-factor F h such that E(H) ∩ E(F h ) = θ, where h: E(G) → [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.

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Acknowledgements

The authors would like to thank the anonymous referees for their kind help and valuable suggestions which led to an improvement of this paper.

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Authors and Affiliations

  1. School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    Si-zhong Zhou

  2. School of Economic and Management, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    Tao Zhang

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  1. Si-zhong Zhou
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  2. Tao Zhang
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Correspondence to Si-zhong Zhou.

Additional information

This work is supported by the National Natural Science Foundation of China (Grant No. 11371009, 11501256, 61503160), and is sponsored by Six Big Talent Peak of Jiangsu Province (Grant No. JY–022) and 333 Project of Jiangsu Province.

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Zhou, Sz., Zhang, T. Some Existence Theorems on All Fractional (g, f)-factors with Prescribed Properties. Acta Math. Appl. Sin. Engl. Ser. 34, 344–350 (2018). https://doi.org/10.1007/s10255-018-0753-y

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  • DOI: https://doi.org/10.1007/s10255-018-0753-y

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