On large matchings and cycles in sparse random graphs

AbstractLet k be a fixed positive integer. A graph H has property Mk if it contains ⌊12k⌋ edge disjoint hamilton cycles plus a further edge disjoint matching which leaves at most one vertex isolated, if k is odd. Let p = c/n, where c is a large enough constant. We show that Gn,p a.s. contains a vertex induced subgraph Hk with property Mk and such that |V (Hk)| = (1 - (1+ϵ(c))ck−1e−c/ (k − 1)!)n, where ϵ(c)→0 as c→∞. In particular this shows that for large c, Gn,p a.s. contains a matching of size 12(1 − (1 + ϵ(c))e−c)n (k = 1) and a cycle of size (1 − (1 + ϵ(c))ce−c)n (k = 2)