Isolated toughness and fractional (a, b, n)-critical graphs. Issue 1 (31st December 2023) - Gao, Wei; Wang, Weifan; Chen, Yaojun
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 δ ( G ) ≥ a + n and I ( G ) > a − 1 + n + 1 n a, b, where n a, b ≥ 2 is an integer satisfies ( 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 ) -critical graphs, Discrete Appl. Math. 322 (2022), 194–202] which established the tight isolated toughness bound for fractional ( k, n ) -critical graphs.