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For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be arandom subgraph of $G$ obtained by taking each edge independently withprobability $p$. We prove that (i) if $p \ge \omega/k$ for a function$\omega=\omega(k)$ that tends to infinity as $k$ does, then $G_p$asymptotically almost surely contains a cycle (and thus a path) of length atleast $(1-o(1))k$, and (ii) if $p \ge (1+o(1))\ln k/k$, then $G_p$asymptotically almost surely contains a path of length at least $k$. Ourtheorems extend classical results on paths and cycles in the binomial randomgraph, obtained by taking $G$ to be the complete graph on $k+1$ vertices.

Long paths and cycles in random subgraphs of graphs with large minimum degree