Sharp Isolated Toughness Bound for Fractional (k, m)-Deleted Graphs

A graph G is a fractional (k, m)-deleted graph if removing any m edges from G, the resulting subgraph still admits a fractional k-factor. Let k ≥ 2 and m ≥ 1 be integers. Denote \(\lfloor{2m \over k}\rfloor^{\ast}=\lfloor{2m \over k}\rfloor\) if \(2m \over k\) is not an integer, and \(\lfloor{2m \over k}\rfloor^{\ast}=\lfloor{2m \over k}\rfloor - 1\) if \(2m \over k\) is an integer. In this paper, we prove that G is a fractional (k, m)-deleted graph if δ(G) ≥ k + m and isolated toughness meets

$$I\left( G \right) > \left\{ {\matrix{{3 - {1 \over m},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {{\text{if}}\,k = 2\,{\text{and}}\,m \ge 3,} \cr {k + {{{{\left\lfloor {{{2m} \over k}} \right\rfloor }^*}} \over {m + 1 - {{\left\lfloor {{{2m} \over k}} \right\rfloor }^*}}},} & {{\text{otherwise}}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr } } \right.$$

Furthermore, we show that the isolated toughness bound is tight.