For a given finite graph G of minimum degree at least k, let Gpβ be a
random subgraph of G obtained by taking each edge independently with
probability p. We prove that (i) if pβ₯Ο/k for a function
Ο=Ο(k) that tends to infinity as k does, then Gpβ
asymptotically almost surely contains a cycle (and thus a path) of length at
least (1βo(1))k, and (ii) if pβ₯(1+o(1))lnk/k, then Gpβ
asymptotically almost surely contains a path of length at least k. Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking G to be the complete graph on k+1 vertices.Comment: 26 page