Tight isolated toughness bound for fractional (k,n)-critical graphs

Isolated toughness of graph is formulated by minimizing the ratio over all with , where is the number of isolated vertices after removing vertex subset from . The previous works reveal that there exist explicit correlations between isolated toughness and fractional critical graphs (a.k.a. the graph admits a fractional factor after deleting given number of vertices). However, among the existing isolated toughness bounds, the term with respect to (the number of removed vertices) is always at least . In this paper, the exactly sharp isolated toughness bound for fractional -critical graph is determined which reveals that the coefficient of term with regard to can be reduced to 1/2. It is well acknowledged that some tight toughness related bounds can reach to the extreme value, while others cannot. We give an explanation why these tight bounds in various settings possess such pattern differences.