Abstract
Let a, b, r be nonnegative integers with \(1\leq{a}\leq{b}\) and \(r\geq2\). Let G be a graph of order n with \(n >\frac{(a+2b)(r(a+b)-2)}{b}\). In this paper, we prove that G is fractional ID-[a, b]-factor-critical if \(\delta(G)\geq\frac{bn}{a+2b}+a(r-1)\) and \(\mid N_{G}(x_{1}) \cup N_{G}(x_{2}) \cup \cdotp \cdotp \cdotp \cup N_{G}(x_{r})\mid\geq\frac{(a+b)n}{a+2b}\) for any independent subset {x1, x2, · · ·, xr} in G. It is a generalization of Zhou et al.’s previous result [Discussiones Mathematicae Graph Theory, 36: 409–418 (2016)] in which r = 2 is discussed. Furthermore, we show that this result is best possible in some sense.
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The authors would like to thank the anonymous referees for their comments on this paper.
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This work was supported by the National Natural Science Foundation of China (Nos. 11371052, 11731002), the Fundamental Research Funds for the Central Universities (Nos. 2016JBM071, 2016JBZ012) and the 111 Project of China (B16002).
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Yuan, Y., Hao, RX. A Neighborhood Union Condition for Fractional ID-[a, b]-factor-critical Graphs. Acta Math. Appl. Sin. Engl. Ser. 34, 775–781 (2018). https://doi.org/10.1007/s10255-018-0786-2
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DOI: https://doi.org/10.1007/s10255-018-0786-2