Abstract
Let k be an integer with k ≥ 1, and let G be a graph. A k-factor of G is a spanning subgraph F of G such that d F (x) = k for each \({x\in V(G)}\) . Let \({h:E(G)\rightarrow[0,1]}\) be a function. If \({\sum_{e\ni x}h(e)=k}\) holds for each \({x\in V(G)}\) , then we call G[F h ] a fractional k-factor of G with indicator function h, where \({F_h=\{e\in E(G): h(e) >0 \}}\) . A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical) if G−I has a fractional k-factor for every independent set I of G. In this paper, we prove that if \({\alpha(G)\leq\frac{4k(\delta(G)-k+1)}{k^{2}+6k+1}}\) , then G is fractional ID-k-factor-critical. The result is best possible in some sense.
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This research was supported by Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (10KJB110003) and Jiangsu University of Science and Technology (2010SL101J), and was sponsored by Qing Lan Project of Jiangsu Province.
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Zhou, S., Xu, L. & Sun, Z. Independence number and minimum degree for fractional ID-k-factor-critical graphs. Aequat. Math. 84, 71–76 (2012). https://doi.org/10.1007/s00010-012-0121-6
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DOI: https://doi.org/10.1007/s00010-012-0121-6