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Isolated toughness and the existence of fractional factors. (Chinese. English summary) Zbl 1016.05049

Let \(G=(V,E)\) be a graph. 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, and \(I(G)=\infty\) otherwise, where \(i(G-S)\) is the number of isolated vertices in \(G-S\). A variation of isolated toughness is defined as \(I'(G)=\min\{|S|/(i(G-S)-1):\;S\subseteq V(G)\), \(i(G-S)\geq 2\}\) if \(G\) is not complete, and \(I'(G)=\infty\) otherwise. This paper discusses the relationships among the isolated toughness, the variation of isolated toughness and fractional factors of a graph, and gives some sufficient conditions for a graph to have a fractional \(1\)-factor or \(2\)-factor. For example, a connected graph \(G\) with order more than two has a fractional \(1\)-factor if it is not complete and \(I'(G)>1\), and has a fractional \(2\)-factor if \(\delta (G)\geq 2\) and \(I(G)\geq 2\).

MSC:

05C40 Connectivity
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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