分数因子存在性问题是图因子理论研究的重要课题, 而孤立韧度是衡量网络易受攻击性的重要参数。作为孤立韧度的唯一变种, $I'(G)$定义为$|S|$ 和$i(G-S)-1$ 的最小比值, 其中$S$ 是满足$i(G-S)\ge2$的顶点子集。该图参数从拓扑结构出发衡量了网络的坚固程度, 并且最近的研究发现其与分数因子之间存在密切的联系。本文给出了一个图存在分数$[a,b]$-因子的$I'(G)$条件, 并且说明该条件是紧的。该结果推广了原来关于分数$k$-因子存在性的$I'(G)$ 紧界。
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.
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