On the number of perfect matchings and Hamilton cycles in \(\varepsilon\)-regular non-bipartite graphs (Q1587507)

Consider a graph on an even number of vertices \(n\). Let \(m\) denote its number of perfect matchings and \(h\) its number of Hamilton cycles. It is shown that if all local densities deviate at most \(\varepsilon\) from the global density \(d\), then \((h/h!)^{1/n}\) and \(2[(n/2)!m/n!]^{2/n}\) both deviate at most \(2\varepsilon\) from \(d\).