Expansion properties of a random regular graph after random vertex deletions

Authors:
Catherine Greenhill;Fred B. Holt;Nicholas Wormald
Affiliations:
School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia;University of Washington, Seattle, WA 98195-4350, USA;Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada N2L 3G1
Venue:
European Journal of Combinatorics
Year:
2008

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Abstract

We investigate the following vertex percolation process. Starting with a random regular graph of constant degree, delete each vertex independently with probability p, where p=n^-^@a and @a=@a(n) is bounded away from 0. We show that a.a.s. the resulting graph has a connected component of size n-o(n) which is an expander, and all other components are trees of bounded size. Sharper results are obtained with extra conditions on @a. These results have an application to the cost of repairing a certain peer-to-peer network after random failures of nodes.