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

A graph G is a fractional $ (a,b,n) $ (a , b , n) -critical graph if removing any n vertices from G, the resulting subgraph still admits a fractional $ [a,b] $ [ a , b ] -factor. In this paper, we determine the exact tight isolated toughness bound for fractional $ (a,b,n) $ (a , b , n) -critical graphs. To be specific, a graph G is fractional $ (a,b,n) $ (a , b , n) -critical if $ \delta (G)\ge a+n $ δ (G) ≥ a + n and $ I(G)>a-1+\frac {n+1}{n_{a,b}} $ I (G) > a − 1 + n + 1 n a , b , where $ n_{a,b}\ge 2 $ n a , b ≥ 2 is an integer satisfies $ (n_{a,b}-1)a\le b\le n_{a,b}a-1 $ (n a , b − 1) a ≤ b ≤ 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) $ (k , n) -critical graphs, Discrete Appl. Math. 322 (2022), 194–202] which established the tight isolated toughness bound for fractional $ (k,n) $ (k , n) -critical graphs. [ABSTRACT FROM AUTHOR]