Random k-out subgraphs

Random sampling a subgraph of a graph is an important algorithmic technique.  Solving some problems on the (smaller) subgraph is naturally faster, and can give either a useful approximate answer, or sometimes even give a result that can be quickly modified to an exact solution, on the original graph.

We study the following simple sampling model and question about it: 

Each vertex of an arbitrary simple graph on n vertices chooses k random incident edges. 

What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph?

We prove that the answer is O(n/k), when k ≥ c log n, for some large enough c.   We conjecture that the same holds for smaller values of k, possibly for any k ≥ 2. Such a result is best possible for any k ≥ 2.

As an application, we use this sampling result to obtain a one-way communication protocol with private randomness for finding a spanning forest of a graph in which each vertex sends only O( \sqrt(n) log n) bits to a referee.  

Based on joint work with Jacob Holm, Valeria King, Mikkel Thorup and Uri Zwick.