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arXiv · 2205.04852

Expansion in Supercritical Random Subgraphs of Expanders and its Consequences

Sahar Diskin·Michael Krivelevich
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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.

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Sahar Diskin, Michael Krivelevich. 2022-05-10. Expansion in Supercritical Random Subgraphs of Expanders and its Consequences. https://arxiv.org/abs/2205.04852

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