We introduce a model of a controlled random graph process. In this model, the
edges of the complete graph Knβ are ordered randomly and then revealed, one
by one, to a player called Builder. He must decide, immediately and
irrevocably, whether to purchase each observed edge. The observation time is
bounded by parameter t, and the total budget of purchased edges is bounded by
parameter b. Builder's goal is to devise a strategy that, with high
probability, allows him to construct a graph of purchased edges possessing a
target graph property P, all within the limitations of observation
time and total budget. We show the following: (a) Builder has a strategy to
achieve minimum degree k at the hitting time for this property by purchasing
at most ckβn edges for an explicit ckβ<k; and a strategy to achieve it
(slightly) after the threshold for minimum degree k by purchasing at most
(1+Ξ΅)kn/2 edges (which is optimal); (b) Builder has a strategy to
create a Hamilton cycle if either tβ₯(1+Ξ΅)nlogn/2 and bβ₯Cn, or tβ₯Cnlogn and bβ₯(1+Ξ΅)n, for some
C=C(Ξ΅); similar results hold for perfect matching; (c) Builder has
a strategy to create a copy of a given k-vertex tree if tβ₯bβ«{(n/t)kβ2,1}, and this is optimal; and (d) For β=2k+1 or
β=2k+2, Builder has a strategy to create a copy of a cycle of length
β if bβ«max{nk+2/tk+1,n/tβ}, and this is optimal.Comment: 20 pages, 2 figure