Packing and counting arbitrary Hamilton cycles in random digraphs

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We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1,…,Ct, each of which is oriented arbitrarily, a digraph ∼(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ∼(n, p) w.h.p. contains (1 ± o(1))n!pn copies of C, provided .