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Isolated toughness and fractional \((g,f)\)-factors of graphs. (English) Zbl 1224.05418

Summary: Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), 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 \(I(G)=| V(G)| -1\). Let \(a\) and \(b\) be positive integers such that \(1\leq a\leq b\), and \(g(x)\) and \(f(x)\) be positive integral-valued functions defined on \(V(G)\) such that \(a\leq g(x)\leq f(x)\leq b\). Let \(h(e)\in [0,1]\) be a function defined on \(E(G)\), \(d^{h}_{G}(x)=\sum _{e\in E_{x}}h(e)\) where \(E_x=\{xy\: xy\in E(G)\}\). Then \(d^{h}_{G}(x)\)) is called the fractional degree of \(x\) in \(G\). We call \(h\) an indicator function if \(g(x)\leq d^{h}_{G}(x)\leq f(x)\) holds for each \(x\in V(G)\). Let \(E^{h}=\{e\: e\in E(G),\;h(e)\neq 0\}\) and \(G_{h}\) be a spanning subgraph of \(G\) such that \(E(G_{h})=E^{h}\). We call \(G_{h}\) a fractional \((g,f)\)-factor. The main results in this paper present some sufficient conditions about isolated toughness for the existence of fractional \((g,f)\)-factors. If \(1=g(x)<f(x)=b\), this condition can be improved and the improved bound is not only sharpness but also a necessary and sufficient condition for a graph to have fractional \([1,b]\)-factor.

MSC:

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