Optimal Construction of Edge-Disjoint Paths in Random Graphs
SIAM Journal on Computing
Scalable secure storage when half the system is faulty
Information and Computation
A proof of alon's second eigenvalue conjecture
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
The emergence of a giant component in random subgraphs of pseudo-random graphs
Random Structures & Algorithms
Percolation on Finite Cayley Graphs
Combinatorics, Probability and Computing
Expansion properties of a random regular graph after random vertex deletions
European Journal of Combinatorics
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We say that a graph G=(V,E) on n vertices is a @b-expander for some constant @b0 if every U@?V of cardinality |U|@?n2 satisfies |N"G(U)|=@b|U| where N"G(U) denotes the neighborhood of U. In this work we explore the process of deleting vertices of a @b-expander independently at random with probability n^-^@a for some constant @a0, and study the properties of the resulting graph. Our main result states that as n tends to infinity, the deletion process performed on a @b-expander graph of bounded degree will result with high probability in a graph composed of a giant component containing n-o(n) vertices that is in itself an expander graph, and constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of (n,d,@l)-graphs, that are such expanders, we compute the values of @a, under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of d-regular graphs with high probability is an expander and meets the additional constraints, this result strengthens a recent result due to Greenhill, Holt and Wormald about vertex percolation on random d-regular graphs. We conclude by showing that performing the above described deletion process on graphs that expand sub-linear sets by an unbounded expansion ratio, with high probability results in a connected expander graph.