Minimum degree edge-disjoint Hamilton cycles in random directed graphs
Asaf Ferber, Adva Mond
TL;DR
This work establishes that for the random digraph $D_{n,p}$ with $p\ge \log^{15}n/n$, the maximum number of edge-disjoint Hamilton cycles equals the minimum of the minimum out- and in-degrees, up to polylogarithmic factors, and provides a randomized algorithm achieving this bound. The authors develop a two-phase approach: first, they build a carefully coupled subdigraph $D'$ containing the requisite number of edge-disjoint 1-factors, then they convert these 1-factors into Hamilton cycles via an online sprinkling technique that exposes edges only as needed while maintaining edge-disjointness. Key technical tools include a bijection between digraphs and bipartite graphs, Gale–Ryser type results for factors, and a phase-structured rotation-extension approach adapted to directed graphs. The result yields a near-tight decomposition of $D_{n,p}$ into $\ig(1+o(1)\big)np$ Hamilton cycles, with the edge-exposure managed to fit within $D_{n,p}$ itself. This advances the understanding of Hamiltonian decompositions in random digraphs and introduces robust online randomness methods for constructing large collections of Hamilton cycles.
Abstract
In this paper we consider the problem of finding ``as many edge-disjoint Hamilton cycles as possible'' in the binomial random digraph $D_{n,p}$. We show that a typical $D_{n,p}$ contains precisely the minimum between the minimum out- and in-degrees many edge-disjoint Hamilton cycles, given that $p\geq \log^{15} n/n$, which is optimal up to a factor of poly$\log n$. Our proof provides a randomized algorithm to generate the cycles and uses a novel idea of generating $D_{n,p}$ in a sophisticated way that enables us to control some key properties, and on an ``online sprinkling'' idea as was introduced by Ferber and Vu.