Packing and counting arbitrary Hamilton cycles in random digraphs

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,…,C_t, each of which is oriented arbitrarily, a digraph ～(n, p) w.h.p. contains edge disjoint copies of C₁,…,C_t, 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 .