Ma, Yinghong; Liu, Guizhen Isolated toughness and the existence of fractional factors. (Chinese. English summary) Zbl 1016.05049 Acta Math. Appl. Sin. 26, No. 1, 133-140 (2003). 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 \(I(G)\geq 2\). Reviewer: Jun-Ming Xu (Hefei) Cited in 6 Documents MSC: 05C40 Connectivity 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) Keywords:fractional factor; isolated toughness PDFBibTeX XMLCite \textit{Y. Ma} and \textit{G. Liu}, Acta Math. Appl. Sin. 26, No. 1, 133--140 (2003; Zbl 1016.05049)