Random k-out subgraph leaves only O(n/k) inter-component edges

09/24/2019

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by   Jacob Holm, et al.

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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> clog 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(√(n)log n) bits to a referee.