Vertex Percolation on Expander Graphs

We say that a graph G = (V,E) on n vertices is a β-expander for some constant β > 0 if every U ⊆ V of cardinality |U | ≤ n 2 satisfies |NG(U)| ≥ β|U | where NG(U) denotes the neighborhood of U . We explore the process of uniformly at random deleting vertices of a β-expander with probability n for some constant α > 0. Our main result implies that as n tends to infinity, the deletion process performed on a β-expander graph of bounded degree will result with high probability in a graph composed of a giant component containing n − o(n) vertices which is itself an expander graph, and small 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, λ)-graphs, which are such expanders, we compute the values of α, 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 hight probability meets all of these constraints, this result strengthens a recent result due to Greenhill, Holt, and Wormald [6] who prove a similar theorem for Gn,d. We conclude by showing that performing the deletion process with the prescribed deletion probability on expander graphs that expand sub-linear sets by an unbounded expansion ratio, with high probability results in an expander graph.