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A neighborhood condition for fractional ID-\([a,b]\)-factor-critical graphs. (English) Zbl 1338.05220

Summary: Let \(G\) be a graph of order \(n\), and let \(a\) and \(b\) be two integers with \(1\leq a\leq b\). Let \(h : E(G)\to [0, 1]\) be a function. If \(a\leq \sum_{e\not\in x}h(e)\leq b\) holds for any \(x\in V(G)\), then we call \(G[F_h]\) a fractional \([a, b]\)-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 \([a, b]\)-factor-critical (in short, fractional ID-\([a, b]\)-factor-critical) if \(G-I\) has a fractional \([a, b]\)-factor for every independent set \(I\) of \(G\). In this paper, it is proved that if \(n\geq\frac{(a+2b)(2a+2b-3)+1}{b}\), \(\delta(G)\geq\frac{bn}{a+2b}+a\) and \(|N_G(x)\cup N_G(y)|\geq\frac{(a+b)n}{a+2b}\) for any two nonadjacent vertices \(x, y\in V(G)\), then \(G\) is fractional ID-\([a, b]\)-factor-critical. Furthermore, it is shown that this result is best possible in some sense.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C72 Fractional graph theory, fuzzy graph theory
05C35 Extremal problems in graph theory
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References:

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