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Fractional factors and isolated toughness of graphs. (English) Zbl 1114.05082

Summary: The isolated toughness of \(G\) is defined as \(I(G)= \min\{|S|/i(G- S): S\subseteq V(G), i(G- S)\geq 2\}\) if \(G\) is not complete. Otherwise, set \(I(G)=\infty\). In this paper, the relationships between the isolated toughness and the existence of fractional \(k\)-factors and fractional \([a, b]\)-factors are given. It is proved that if \(\delta(G)\geq k\) and \(I(G)\geq k\), then \(G\) has a fractional \(k\)-factor; if \(\delta(G)\geq I(G)\geq a- 1+ a/b\), then \(G\) has a fractional \([a, b]\)-factor where \(a< b\). Furthermore, it is showed that the results in this paper are best possible in some sense.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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