The asymptotic connectivity of labelled regular graphs

It is shown that almost all labelled r -regular graphs are r -connected, for any fixed r ≥3. This is an instance of a more general result which applies to labelled locally restricted graphs whose points have degrees lying between r and R , where r and R are fixed, 3≤ r ≤ R . The proof employs a known asymptotic formula for the numbers of these locally restricted graphs. It is also demonstrated that for fixed k >0 and r ≥3, almost all labelled r -regular graphs with girth at least [ k /( r −2)] are cyclically k -connected. This provides an asymptotic formula for the number of labelled cyclically k -connected r -regular graphs with p points, for fixed k and r .