Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, October 21, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Michael Krivelevich (Tel Aviv University)

Contagious sets in random graphs

Consider the following activation process in graphs: a vertex is active either if it belongs to a set of initially activated vertices, or at some point it has at least 2 active neighbors. (In perhaps more commonly used terms, this is the so called bootstrap percolation with a threshold parameter $r=2$.)

A CONTAGIOUS SET in a graph $G$ is a set whose activation results with the entire graph being active. Let $m(G,2)$ be the minimum size of a contagious set in $G$.

Recently, in a joint work with Uriel Feige and Daniel Reichman, we showed that for the binomial random graph $G\sim G(n,p)$, with $p=d/n$ and $1 \ll d \ll n^{1/2}$, the value of the parameter $m(G,2)$ is typically of the asymptotic order $n/(d^2\log d)$.

In this talk I will discuss this result, as well as some prior/relevant papers.

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