Counting Hamilton cycles in sparse random directed graphs
نویسندگان
چکیده
Let D(n, p) be the random directed graph on n vertices where each of the n(n− 1) possible arcs is present independently with probability p. It is known that if p ≥ (log n+ ω(1))/n then D(n, p) typically has a directed Hamilton cycle, and this is best possible. We show that under the same condition, the number of directed Hamilton cycles in D(n, p) is typically n!(p(1 + o(1))) . We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically n!(log n/n(1 + o(1))) directed Hamilton cycles.
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