Avoiding a giant component

Avoiding a Giant Component

Avoiding a giant component

A phase transition for avoiding a giant component

Let c be a constant and (e1, f1), (e2, f2), . . . , (ecn, fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2 then whp every graph which contains at least one edge from each ordered pair (ei, fi) has a component of size Ω(n) and if c < c2 then whp there is a graph conta...

Creating a Giant Component

Let c be a constant and (e1, f1), (e2, f2), . . . , (ecn, fcn) be a sequence of ordered pairs of edges on vertex set [n] chosen uniformly and independently at random. Let A be an algorithm for the on-line choice of one edge from each presented pair, and for i = 1, . . . , cn let GA(i) be the graph on vertex set [n] consisting of the first i edges chosen by A. We prove that all algorithms in a c...

The Giant Component

But, we will consider a fancier approach to sampling that yields the same distribution. In particular, we will create a collection of independent random variables Xi,j for all i 6= j. This is twice as many variables as we need, so we will only use one from each pair (Xi,j , Xj,i). We will then choose pairs (i, j) in some order, choose only one of Xi,j or Xj,i to examine, and then use the variab...