Site Percolation on Pseudo-Random Graphs

TL;DR

This paper investigates vertex percolation on pseudo-random $d$-regular graphs, establishing phase transition behavior, asymptotic sizes of the largest component, and properties in both supercritical and subcritical regimes.

Contribution

It provides new precise asymptotics for the largest component and analyzes additional properties like cycles and expansion in pseudo-random graphs.

Findings

Identifies the phase transition at p=1/d.

Determines the asymptotic size of the largest component in the supercritical regime.

Strengthens bounds on component sizes in the subcritical regime.

Abstract

We consider vertex percolation on pseudo-random $d-$regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in $\frac{n}{d}$) sized component, at $p=\frac{1}{d}$. In the supercritical regime, our main result recovers the sharp asymptotic of the size of the largest component, and shows that all other components are typically much smaller. Furthermore, we consider other typical properties of the largest component such as the number of edges, existence of a long cycle and expansion. In the subcritical regime, we strengthen the upper bound on the likely component size.