Edge disjoint Hamilton cycles in random digraphs of constant minimum degree

TL;DR

This paper proves that random directed graphs with sufficiently large minimum in- and out-degree contain multiple edge-disjoint Hamilton cycles with high probability.

Contribution

It establishes the existence of multiple edge-disjoint Hamilton cycles in random digraphs conditioned on minimum degree constraints.

Findings

Random digraphs with minimum degree at least k+1 contain k edge-disjoint Hamilton cycles.

High probability of existence of these cycles in large graphs.

Results hold for graphs with m=cn edges, c above a certain threshold.

Abstract

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and $c\ge c_k$, where $c_k$ is a sufficiently large constant. We prove that w.h.p. $D_{n,m}^{(\delta\geq k+1)}$ contains $k$ edge disjoint Hamilton cycles.