Random regular graphs of high degree (Q2746213)

For a positive integer-valued function \(d= d(n)\) the model \(G_{n,d}\) of random regular graphs is defined to consist of all regular graphs on \(n\) vertices of degree \(d\) with uniform probability distribution. This paper provides a range of results on asymptotic properties of \(G_{n,d}\) as \(n\to\infty\) when \(d(n)\) grows more quickly than \(\sqrt n\). Its aim is to fill the gap in the range of possible values of \(d\) since, as the authors claim, ``at the present time nothing has really been published for \(d\) bigger than \(\sqrt n\).'' The properties of \(G_{n,d}\) studied in the paper include connectivity, Hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, among others.