Tight isolated toughness bound for fractional (a,b,m)-deleted graphs

A graph [Formula: see text] is a fractional [Formula: see text]-deleted graph if deleting any [Formula: see text] edges from [Formula: see text], the resulting subgraph still admits a fractional [Formula: see text]-factor. As an exclusive graph-based parameter, isolated toughness and its variant are utilized to measure the vulnerability of networks which are modelled by graph topology structures. Early studies have shown the inherent connection between isolated toughness and the existence of fractional factors in specific settings. This paper provides a theoretical perspective on the sufficient conditions for fractional [Formula: see text]-deleted graphs with respect to isolated toughness and its variant. The sharpness of the given isolated toughness bounds is explained by counterexamples.