Packing and counting arbitrary Hamilton cycles in random digraphs

© 2018 Wiley Periodicals, Inc. We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in D(n, p) for nearly optimal p (up to a logcn factor). In particular, we show that given t = (1 - o(1))np Hamilton cycles C-1, horizontal ellipsis ,C-t, each of which is oriented arbitrarily, a digraph D similar to D(n, p) w.h.p. contains edge disjoint copies of C-1, horizontal ellipsis ,C-t, provided p=omega(log3n/n). We also show that given an arbitrarily oriented n-vertex cycle C, a random digraph D similar to D(n, p) w.h.p. contains (1 +/- o(1))n!p(n) copies of C, provided p >= log1+o(1)n/n.