#### Conference abstracts

Session B2 - Graph Theory and Combinatorics

July 13, 16:00 ~ 16:25 - Room B7

## Critical percolation on random regular graphs

### Guillem Perarnau

### University of Birmingham, United Kingdom - g.perarnau@bham.ac.uk

We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices at the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known results for fixed $d\geq 3$ and for $d=n-1$, confirming a prediction of Nachmias and Peres on a question of Benjamini. In contrast to previous approaches, our proof is based on an application of the switching method.

Joint work with Felix Joos (University of Birmingham).