# The Unit Acquisition Number of Binomial Random Graphs

Let *G* be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex *u *to a neighbouring vertex *v* can be moved, provided that the weight on *v* is at least as large as the weight on *u*. The unit acquisition number of *G*, denoted by *a*_{u}*(G)*, is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdõs-Rényi random graph process (*G(n,m)*)^{N}* *_{m=0}, where N =(^{n} _{2}).We show that asymptotically almost surely *a*_{u}*(G)(n,m))* = 1 right at the time step the random graph process creates a connected graph. Since trivially *a*_{u}*(G)(n,m))* ≥ 2 if the graphs is disconnected, the result holds in the strongest possible sense.