Vertex-minor universality of a random graph

TL;DR

This paper proves that random graphs G(n,p) are highly vertex-minor universal for a wide range of p, confirming a conjecture and extending previous results with high probability bounds.

Contribution

It confirms a conjecture that G(n,p) graphs are vertex-minor universal for all p in a broad range, up to a logarithmic factor, with high probability.

Findings

G(n,p) is (p\u2212) () vertex-minor universal with high probability.

The result holds for ( ())  p  1/2.

Confirms the conjecture for the entire range of p from (1/) to 1/2.

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 $\Omega(\sqrt{n})$-vertex-minor universal. Furthermore, they conjectured that with high probability $G(n,p)$ and $G(n,1-p)$ are $\Omega(p\sqrt{n})$-vertex-minor universal for all $\omega(1/\sqrt{n})\le p\le 1/2$. In this short note, we confirm this conjecture up to an extra logarithm factor and show that this is true with probability $1-2^{-\Omega(p^2n)}$ if $\Omega(\log n/\sqrt{n})\le p\le 1/2$. 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.