Almost all graphs are vertex-minor universal

TL;DR

This paper proves that random graphs are highly vertex-minor universal, enabling the realization of any smaller graph as a vertex-minor, with significant implications for quantum networks and matroid theory.

Contribution

It establishes that random graphs are (.911 db7 d) ext{-vertex-minor universal, introduces a vertex-minor Ramsey number, and extends results to bipartite pivot-minors and binary matroids.

Findings

Random graphs are (.911 db7 d) ext{-vertex-minor universal with high probability.

The vertex-minor Ramsey number } R_{ ext{vm}}(k) ext{ is between } \u0010(k^2) ext{ and } 2^k - 1.

Applications to quantum stabilizer states and universality in random binary matroids.

Abstract

Answering a question of Claudet, we prove that the uniformly random graph $G\sim \mathbb G(n, 1/2)$ is $\Omega(\sqrt n)$-vertex-minor universal with high probability. That is, for some constant $\alpha\approx 0.911$, any graph on any $\alpha\sqrt n$ specified vertices of $G$ can be obtained as a vertex-minor of $G$. This has direct implications for quantum communications networks: an $n$-vertex $k$-vertex-minor universal graph corresponds to an $n$-qubit $k$-stabilizer universal graph state, which has the property that one can induce any stabilizer state on any $k$ qubits using only local operations and classical communications. We further employ our methods in two other contexts. We obtain a bipartite pivot-minor version of our main result, and we use it to derive a universality statement for minors in random binary matroids. We also introduce the vertex-minor Ramsey number $R_{\mathrm{vm}}(k)$ to be the smallest value $n$ such that every $n$-vertex graph contains an independent set of size $k$ as a vertex-minor. Supported by our main result, we conjecture that $R_{\mathrm{vm}}(k)$ is polynomial in $k$. We prove $\Omega(k^2) \leq R_{\mathrm{vm}}(k) \leq 2^k - 1$.