An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

<p>We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph . In this model <em>G</em> is drawn uniformly from graphs with vertex set [<em>n</em>], <em>m</em> edges and minimum degree at least three. We focus on the case where <em>m</em> = <em>cn</em> for constant <em>c</em>. If <em>c</em> is sufficiently large then our algorithm runs in time and succeeds w.h.p.</p

A scaling limit for the length of the longest cycle in a sparse random graph

We discuss the length L-c,L-n of the longest cycle in a sparse random graph G(n,p), p = c/n, c constant. We show that for large c there exists a function f (c) such that L-c,L-n/n -> f (c) a.s. The function f (c) = 1 - Sigma(infinity)(k=1) p(k)(c)e(-kc) where pk(c) is a polynomial in c. We are only able to explicitly give the values p(1), p(2), although we could in principle compute any p(k). We see immediately that the length of the longest path is also asymptotic to f (c)n w.h.p