Random regular graphs with edge faults: Expansion through cores

Abstract

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1−p ). In particular, we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree d. Here we deal with expansion properties of faulty random regular graphs and show: For fixed d⩾42 and p = κ/d, κ⩾20 , a random regular graph with fault probability f = 1−p contains a linear-size subgraph which is an expander almost surely. This subgraph can be found by a simple linear-time algorithm.