Isolated toughness and fractional (a,b,n)-critical graphs

Abstract

A graph G is a fractional $(a,b,n)$-critical graph if removing any n vertices from G, the resulting subgraph still admits a fractional $[a,b]$-factor. In this paper, we determine the exact tight isolated toughness bound for fractional $(a,b,n)$-critical graphs. To be specific, a graph G is fractional $(a,b,n)$-critical if $\delta (G)\ge a+n$ and $I(G)>a-1+\frac{n+1}{n_{a,b}}$, where $n_{a,b}\ge 2$ is an integer satisfies $(n_{a,b}-1)a\le b\le n_{a,b}a-1$. Furthermore, the sharpness of bounds is showcased by counterexamples. Our contribution improves a result from [W. Gao, W. Wang, and Y. Chen, Tight isolated toughness bound for fractional $(k,n)$-critical graphs, Discrete Appl. Math. 322 (2022), 194–202] which established the tight isolated toughness bound for fractional $(k,n)$-critical graphs.

Keywords:

• Graph
• isolated toughness
• fractional $[a,b]$-factor
• fractional $(a,b,n)$-critical graph

AMS:

• 05C70

Acknowledgments

We thank the reviewers for their constructive comments in improving the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been partially supported by National Natural Science Foundation of China [grant numbers 12161094, 12031018, 12161141003, 11931006].