Optimal packings of Hamilton cycles in sparse random graphs

TL;DR

This paper proves that in sparse random graphs with certain edge probabilities, there almost surely exist many edge-disjoint Hamilton cycles, extending understanding of Hamiltonian structures in random graphs.

Contribution

It establishes the existence of a large collection of edge-disjoint Hamilton cycles in sparse random graphs within a specific probability range.

Findings

Existence of loor{rac{ ext{minimum degree}}{2}} edge-disjoint Hamilton cycles in G(n,p)

Results hold for p between rac{ extlog n}{n} and n^{-1+ extepsilon}

Asymptotic almost sure existence of these Hamilton cycles

Abstract

We prove that there exists a positive constant \epsilon such that if \log n / n \le p \le n^{-1+\epsilon}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor \delta(G)/2 \rfloor edge-disjoint Hamilton cycles.