The existence of fractional factors in specific settings is an important topic of graph factor theory, and isolated toughness is an important parameter to measure the vulnerability of networks. As the unique variant of isolation toughness, $I'(G)$ is defined as the minimum ratio of $|S|$ and $i(G-S)-1$, where $S$ is the subset of vertices that satisfies $i(G-S)\ge2$. This parameter measures the robustness of the network from the perspective of topology, and recent research reveals that it is closely related to the fractional factor. In this paper, we give an $I'(G)$ condition for the existence of fractional $[a,b]$-factors in a graph, and show that the condition is sharp by counterexample. This result extends the original $I'(G)$ tight bound on the existence of the fractional $k$-factor.
[1] Bondy J A, Mutry U S R. Graph Theory [M]. Berlin: Springer, 2008.
[2] 杨景波，马英红，刘桂真.图的分数（g，f)-因子[J].高校应用数学学报A辑，2001,16(4)：385-390.
[3] 马英红，刘桂真.图的孤立韧度与分数因子的存在性[J].应用数学学报，2003，26(1):133-140.
[4] Ma Y H, Liu G Z. Fractional factors and isolated toughness of graphs [J]. 应用数学, 2006, 19(1): 188-194.
[5] Gao W, Wang W F, Chen Y J. Tight isolated toughness bound for fractional (k; n)-critical graphs [J]. Discrete Applied Mathematics, 2022, 322: 194-202.
[6] He Z Y, Liang L, Gao W. Isolated toughness variant and fractional k-factor [J]. RAIROOperations Research, 2022, 56(5): 3675-3688.
[7] Liu G Z, Zhang L J. Fractional (g; f)-factors of graph [J]. Acta Mathematica Scientia, 2001, 21B(4): 541-545.
[8] Liu G Z, Zhang L J. Toughness and the existence of fractional k-factors of graphs [J]. Discrete Mathematics, 2008, 308: 1741-1748.
