Expansion in Supercritical Random Subgraphs of Expanders and its Consequences

Abstract

In 2004, Frieze, Krivelevich and Martin [17] established the emergence of a giant component in random subgraphs of pseudo-random graphs. We study several typical properties of the giant component, most notably its expansion characteristics. We establish an asymptotic vertex expansion of connected sets in the giant by a factor of $\tilde{O}\left({\epsilon }^2\right)$. From these expansion properties, we derive that the diameter of the giant is typically $O_{\epsilon }\left(\log n\right)$, and that the mixing time of a lazy random walk on the giant is asymptotically $O_{\epsilon }\left({\log }^{2}n\right)$. We also show similar asymptotic expansion properties of (not necessarily connected) linear sized subsets in the giant, and the typical existence of a large expander as a subgraph.