Arboricity and spanning-tree packing in random graphs with an application to load balancing

TL;DR

This paper analyzes the arboricity and spanning-tree packing in Erdős-Rényi random graphs, providing precise thresholds and concentration results, with applications to load balancing and network design.

Contribution

It offers exact characterizations and threshold values for spanning trees and arboricity in random graphs, extending to sequential edge addition and load balancing applications.

Findings

T equals the minimum of delta and floor(m/(n-1)) with high probability

Identifies a sharp threshold for p where T and A change behavior

Provides a concentration result for arboricity below the threshold

Abstract

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erdos-Renyi random graph G(n,p). For all p(n) in [0,1], we show that, with high probability, T is precisely the minimum between delta and floor(m/(n-1)), where delta is the smallest degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for p such that: above this threshold, T equals floor(m/(n-1)) and A equals ceiling(m/(n-1)); and below this threshold, T equals delta, and we give a two-value concentration result for the arboricity A in that range. Finally, we include a stronger version of these results in the context of the random graph process where the edges are sequentially added one by one. A direct application of our result gives a sharp threshold for the maximum load being at most k in the two-choice load balancing problem, where k goes to infinity.