Neighborhood unions and factor critical graphs (Q1301846)

A graph \(G\) is \(n\)-factor-critical if \(G - S\) has a perfect matching for any set \(S\) of \(n\) vertices of \(G\). A sufficient condition in terms of generalized degrees of independent sets is given that implies a graph is \(n\)-factor-critical. More specifically, it is shown that if \(G\) is a \(k\)-connected graph of order \(p\), \(n\) is an integer with \(0 \leq n \leq k\) and \(p \equiv n\) (mod \(2\)), \(\alpha\) is a real number with \(1/2 \leq \alpha \leq 1\), and \(|N_G(A)|> \alpha(p - 2k + n - 2) + k\) for every independent set \(A\) of \(G\) with \(|A|= \lfloor \alpha(k - n + 2) \rfloor\), then \(G\) is \(n\)-factor-critical.