The issues of ruggedness and vulnerability are cruxes in network security research, which must be considered during the network designing phase. Parameters such as toughness, isolated toughness, and binding number characterize the vulnerable of the network from the structure of networks. The path factor, a special case of the generalized $\mathrm{\mathscr{H}}$‐factor, measures the feasibility of data transmission in networks. Recent advances have been obtained to show that there is an inevitable connection between the vulnerability parameters of the network and the existence of path factors, while we found that some existing theoretical results are not tight and there is still a long way for further improvement. In view of graph theory approaches, this paper mainly contributes to determine the sharp bounds of toughness, isolated toughness, and binding number for the existence of path factor in different settings, and therefore solve the open problems left unsolved in previous articles.

中文翻译：

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网络漏洞参数设置中存在路径因素的严格界限

坚固性和脆弱性问题是网络安全研究的关键，在网络设计阶段必须考虑这些问题。诸如韧性，孤立韧性和绑定数之类的参数从网络结构上表征了网络的脆弱性。路径因素，广义的特殊情况$\mathrm{\mathscr{H}}$因素，衡量网络中数据传输的可行性。最近的进展表明，网络的脆弱性参数和路径因素的存在之间存在必然的联系，而我们发现一些现有的理论结果并不严格，并且还有很长的路要走。鉴于图论方法，本文主要为确定不同设置下路径因子的存在而确定的韧性，孤立韧性和结合数的尖锐界限，从而解决了以前文章中未解决的开放问题。