Packing and counting arbitrary Hamilton cycles in random digraphs - Ferber, A; Long, E (Wiley, 2018)
We prove packing and counting theorems for arbitrarily ori-ented Hamilton cycles in (n, p) for nearly optimal p (uptoalogcn factor). In particular, we show that given t =(1−o(1))npHamilton cycles C1, … , Ct, each of which is oriented arbi-trarily, a digraph D ∼ (n, p) w.h.p. contains edge disjointcopies of C1, … , Ct, provided p = (log3n∕n).Wealsoshowthat given an arbitrarily oriented n-vertex cycle C, a randomdigraph D ∼ (n, p) w.h.p. contains (1 ± o(1))n!pncopies ofC, provided p ≥ log1+o(1)n∕n.