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

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