- Series
- Combinatorics Seminar
- Time
- Friday, September 20, 2024 - 3:15pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Sam Spiro – Rutgers University – sas703@scarletmail.rutgers.edu – https://samspiro.xyz/
- Organizer
- Jiaxi Nie

Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a quasikernel if $Q$ is an independent set and if every vertex of $D$ can be reached from $Q$ by a path of length at most 2. The Small Quasikernel Conjecture of P.L. Erdős and Székely from 1976 states that every $n$-vertex source-free digraph $D$ contains a quasikernel of size at most $\frac{1}{2}n$. Despite being posed nearly 50 years ago, very little is known about this conjecture, with the only non-trivial upper bound of $n-\frac{1}{4}\sqrt{n\log n}$ being proven recently by ourself. We discuss this result together with a number of other related results and open problems around the Small Quasikernel Conjecture.