3 research outputs found
Fast construction on a restricted budget
We introduce a model of a controlled random graph process. In this model, the
edges of the complete graph 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 , and the total budget of purchased edges is bounded by
parameter . 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 , all within the limitations of observation
time and total budget. We show the following: (a) Builder has a strategy to
achieve minimum degree at the hitting time for this property by purchasing
at most edges for an explicit ; and a strategy to achieve it
(slightly) after the threshold for minimum degree by purchasing at most
edges (which is optimal); (b) Builder has a strategy to
create a Hamilton cycle if either and , or and , for some
; similar results hold for perfect matching; (c) Builder has
a strategy to create a copy of a given -vertex tree if , and this is optimal; and (d) For or
, Builder has a strategy to create a copy of a cycle of length
if , and this is optimal.Comment: 20 pages, 2 figure
Creating Small Subgraphs in Achlioptas Processes with Growing Parameter
Abstract. We study the problem of creating a copy of some fixed graph H in the Achlioptas process on n vertices with parameter r, where r = r(n) is a growing function of n. We prove general upper and lower bounds on the threshold of this problem, and derive exact threshold functions for the case where H is a tree, a cycle, or the complete graph on four vertices. 1