Note on packing of edge-disjoint spanning trees in sparse random graphs

TL;DR

This paper investigates the conditions under which the maximum number of edge-disjoint spanning trees in sparse random graphs equals the minimum degree, improving previous bounds and providing new probabilistic thresholds.

Contribution

It establishes new bounds on the edge probability p for which the spanning tree packing number equals or is less than the minimum degree in G(n,p), refining earlier results.

Findings

For p between (log n + ω(1))/n and (1.1 log n)/n, the packing number equals the minimum degree almost surely.

For p ≥ (51 log n)/n, the packing number is almost surely less than the minimum degree.

The results extend and improve the bounds for the relation between spanning tree packing number and minimum degree in sparse random graphs.

Abstract

The \emph{spanning tree packing number} of a graph $G$ is the maximum number of edge-disjoint spanning trees contained in $G$. Let $k\geq 1$ be a fixed integer. Palmer and Spencer proved that in almost every random graph process, the hitting time for having $k$ edge-disjoint spanning trees equals the hitting time for having minimum degree $k$. In this paper, we prove that for any $p$ such that $(\log n+\omega(1))/n\leq p\leq (1.1\log n)/n$, almost surely the random graph $G(n,p)$ satisfies that the spanning tree packing number is equal to the minimum degree. Note that this bound for $p$ will allow the minimum degree to be a function of $n$, and in this sense we improve the result of Palmer and Spencer. Moreover, we also obtain that for any $p$ such that $p\geq (51\log n)/n$, almost surely the random graph $G(n,p)$ satisfies that the spanning tree packing number is less than the minimum degree.