Covering Random Digraphs with Hamilton Cycles

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.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 2.1: Chernoff Bound
• Theorem 2.2: JRLrandomgraphs, Corollary 2.4
• Theorem 2.3: bergefourniervizing
• Theorem 2.4: Hall's Theorem
• Corollary 2.5
• Lemma 2.6
• Lemma 2.7: OreStudies63
• Definition 2.8
• Lemma 2.9