Toughness of graphs and \([2,b]\)-factors (Q1334935)

It is proved that if a graph \(G\) has order at least 3, and \(b>2\), and for the toughness of \(G\) it holds \(\text{tough} (G) \geq 1 + {1 \over b}\), then \(G\) has a \([2,b]\)-factor. Moreover, it is shown that the thoughness condition in the above theorem cannot be replaced by \(\text{tough} (G) \geq 1 + {1 \over b} - \Theta\), where \(\Theta\) is any positive real number.