Vertex-minor universality of a random graph

Abstract

Given a graph $G$ and a vertex $v\in V(G)$, a local complementation at $v$ on $G$ is an operation that replaces the induced graph on the neighborhood of $v$ by its complement. A graph $H$ is a vertex-minor if $H$ can be obtained from $G$ by a sequence of vertex deletions and local complementation. A graph is said to be $k$-vertex-minor universal if it contains every $k$-vertex graph on any $k$-subset of vertices as a vertex minor. Previously, Ascoli--Fredrickson--Fredrickson--McFarland--Post proved that with high probability $G(n,1/2)$ is $Ω(\sqrt{n})$-vertex-minor universal. Furthermore, they conjectured that with high probability $G(n,p)$ and $G(n,1-p)$ are $Ω(p\sqrt{n})$-vertex-minor universal for all $ω(1/\sqrt{n})\le p\le 1/2$. In this short note, we confirm this conjecture up to an extra logarithm factor and show that this is true with probability $1-2^{-Ω(p^2n)}$ if $Ω(\log n/\sqrt{n})\le p\le 1/2$. Together with a complementary result which applies to the regime where $1/\sqrt{n}\le p\le n^{-1/3}$ produced by an internal model at OpenAI, the conjecture is fully confirmed.