Covering Random Digraphs with Hamilton Cycles

TL;DR

This paper proves that for a wide range of probabilities, the random directed graph can be covered by the minimum number of Hamilton cycles, matching the maximum degree, which is optimal.

Contribution

It establishes the optimal Hamilton cycle covering for random digraphs in a broad probability range, improving understanding of their structural properties.

Findings

Random digraphs with p between log^{20}(n)/n and 1/2 can be covered by Hamilton cycles.

The number of Hamilton cycles needed equals the maximum in-degree or out-degree.

The bound on p is nearly optimal, confirming the robustness of the result.

Abstract

A covering of a digraph $D$ by Hamilton cycles is a collection of directed Hamilton cycles (not necessarily edge-disjoint) that together cover all the edges of $D$. We prove that for $1/2 \geq p\geq \frac{\log^{20} n}{n}$, the random digraph $D_{n,p}$ typically admits an optimal Hamilton cycle covering. Specifically, the edges of $D_{n,p}$ can be covered by a family of $t$ Hamilton cycles, where $t$ is the maximum of the the in-degree and out-degree of the vertices in $D_{n,p}$. Notably, $t$ is the best possible bound, and our assumption on $p$ is optimal up to a polylogarithmic factor.