The probability of selecting $k$ edge-disjoint Hamilton cycles in the complete graph

TL;DR

This paper calculates the probability that $k$ randomly chosen Hamilton cycles in a complete graph are edge-disjoint, extending previous results for the case $k=2$ to larger $k$ values.

Contribution

It provides an asymptotic probability estimate for the edge-disjointness of multiple random Hamilton cycles in complete graphs for small $k$.

Findings

Probability of $k$ edge-disjoint Hamilton cycles is approximately $e^{-2\binom{k}{2}}$ for $k = o(n^{1/100})$.

Extends Robbins' result from $k=2$ to larger $k$ values.

Shows the probability tends to zero rapidly as $k$ increases.

Abstract

Let $H_1,\dots,H_k$ be Hamilton cycles in $K_n$, chosen independently and uniformly at random. We show, for $k = o(n^{1/100})$, that the probability of $H_1,\dots,H_k$ being edge-disjoint is $(1+o(1))e^{-2\binom{k}{2}}$. This extends a corresponding estimate obtained by Robbins in the case $k=2$.