No abstract

We discuss the length 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,n /n → f (c) a.s. The function f (c) = 1 − ∞ k=1 p k (c)e −kc where p k 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.

In this paper, we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular, we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge‐disjoint rainbow‐Hamilton cycles w.h.p. We also ask when, in the resultant graph, every pair of vertices is connected by a rainbow path w.h.p.

In this paper we consider the existence of Hamilton cycles in the random graph G = G ≥3 n,m. This random graph is chosen uniformly from  ≥3 n,m , the set of graphs with vertex set [n], m edges and minimum degree at least 3. Our ultimate goal is to prove that if m = cn and c > 3∕2 is constant then G is Hamiltonian w.h.p. In Frieze (2014), the second author showed that c ≥ 10 is sufficient for this and in this paper we reduce the lower bound to c > 2.662 …. This new lower bound is the same lower bound found in Frieze and Pittel (2013) for the expansion of so-called Pósa sets.

We prove that for k + 1 ≥ 3 and c > (k + 1)/2 w.h.p. the random graph on n vertices, cn edges and minimum degree k + 1 contains a (near) perfect k-matching. As an immediate consequence we get that w.h.p. the (k +1)-core of G n,p , if non empty, spans a (near) spanning kregular subgraph. This improves upon a result of Chan and Molloy [6] and completely resolves a conjecture of Bollobás, Kim and Verstraëte [5]. In addition, we show that w.h.p. such a subgraph can be found in linear time. A substantial element of the proof is the analysis of a randomized algorithm for finding k-matchings in random graphs with minimum degree k + 1.