Packing and counting arbitrary Hamilton cycles in random digraphs

TL;DR

This paper establishes near-optimal conditions for packing and counting arbitrarily oriented Hamilton cycles in random directed graphs, advancing understanding of their structure and abundance.

Contribution

It proves new packing and counting theorems for arbitrarily oriented Hamilton cycles in random digraphs at nearly optimal edge probability thresholds.

Findings

Edge-disjoint packing of nearly all Hamilton cycles with arbitrary orientation at p = ω(log^3 n / n)

Counting of Hamilton cycles with arbitrary orientation at p ≥ log^{1+o(1)} n / n

Results hold with high probability in the random digraph model D(n,p)

Abstract

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$, each of which is oriented arbitrarily, a digraph $D \sim {\cal D}(n,p)$ w.h.p. contains edge disjoint copies of $C_1,\ldots ,C_t$, provided $p=\omega(\log ^3 n/n)$. We also show that given an arbitrarily oriented $n$-vertex cycle $C$, a random digraph $D \sim {\cal D}(n,p)$ w.h.p. contains $(1\pm o(1))n!p^n$ copies of $C$, provided $p \geq \log ^{1 + o(1)}n/n$.