Minimum degree, independence number and regular factors (Q1906860)

The author shows that if for a graph \(G\), \(\delta(G)> \alpha(G)\), then \(G\) has a 2-factor and goes on to describe those graphs in which \(\delta(G)= \alpha(G)\) but \(G\) has no 2-factor. (Note that \(\delta(G)\) denotes the minimum degree in \(G\), and \(\alpha(G)\) the independence number.) In the case of \(k\)-factors, a lower bound on \(\kappa(G)\) (the connectivity of \(G\)) in terms of \(\alpha(G)\) and \(k\) which guarantees a \(k\)-factor is given.