Site Percolation on Pseudo-Random Graphs

• Main Authors: Diskin, Sahar, Krivelevich, Michael
• Format: Journal Article
• Language: English
• Published: 28-07-2021
• Subjects: Mathematics - Combinatorics; Mathematics - Probability
• DOI: 10.48550/arxiv.2107.13326

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.