  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.
