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Almost all graphs are vertex-minor universal

Ruben Ascoli, Bryce Frederickson, Sarah Frederickson, Caleb McFarland, Logan Post

TL;DR

This work shows that a uniformly random graph $G\sim \mathbb{G}(n,1/2)$ is $\Omega(\sqrt{n})$-vertex-minor universal with high probability, establishing that for $k=\alpha\sqrt{n}$ (with $\alpha\approx 0.911$) every graph on $k$ vertices appears as a vertex-minor of $G$. The authors construct and analyze two graph-space random walks to prove rapid mixing to the uniform distribution, and they deploy a reordering lemma and a second-moment argument to bound correlations between many potential vertex-minor occurrences. The results extend to pivot-minors and to minors of random binary matroids, yielding a bipartite pivot-minor universality and linking vertex-minor universality to matroid theory; they also define and bound the vertex-minor Ramsey number $R_{vm}(k)$. Together these findings illuminate probabilistic methods for universal quantum graph states and have implications for quantum communications networks, where stabilizer universality can be realized via vertex-minor operations on random graph states.

Abstract

Answering a question of Claudet, we prove that the uniformly random graph $G\sim \mathbb G(n, 1/2)$ is $Ω(\sqrt n)$-vertex-minor universal with high probability. That is, for some constant $α\approx 0.911$, any graph on any $α\sqrt n$ 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 $Ω(k^2) \leq R_{\mathrm{vm}}(k) \leq 2^k - 1$.

Almost all graphs are vertex-minor universal

TL;DR

This work shows that a uniformly random graph is -vertex-minor universal with high probability, establishing that for (with ) every graph on vertices appears as a vertex-minor of . The authors construct and analyze two graph-space random walks to prove rapid mixing to the uniform distribution, and they deploy a reordering lemma and a second-moment argument to bound correlations between many potential vertex-minor occurrences. The results extend to pivot-minors and to minors of random binary matroids, yielding a bipartite pivot-minor universality and linking vertex-minor universality to matroid theory; they also define and bound the vertex-minor Ramsey number . Together these findings illuminate probabilistic methods for universal quantum graph states and have implications for quantum communications networks, where stabilizer universality can be realized via vertex-minor operations on random graph states.

Abstract

Answering a question of Claudet, we prove that the uniformly random graph is -vertex-minor universal with high probability. That is, for some constant , any graph on any specified vertices of can be obtained as a vertex-minor of . This has direct implications for quantum communications networks: an -vertex -vertex-minor universal graph corresponds to an -qubit -stabilizer universal graph state, which has the property that one can induce any stabilizer state on any 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 to be the smallest value such that every -vertex graph contains an independent set of size as a vertex-minor. Supported by our main result, we conjecture that is polynomial in . We prove .
Paper Structure (14 sections, 43 theorems, 66 equations)

This paper contains 14 sections, 43 theorems, 66 equations.

Table of Contents

  1. Background and main results
  2. Preliminaries and proof overview
  3. Vertex-minor preliminaries
  4. Probability preliminaries
  5. Proof overview
  6. Proof of vertex-minor universality of the random graph
  7. Random walks on the space of graphs
  8. Proof of \ref{['thm:vm_universal']}
  9. Proof of reordering \ref{['lem:minor-reordering']}
  10. Universality for pivot-minors and binary matroids
  11. Reduction from binary matroids to bipartite pivot-minors
  12. Bipartite pivot-minor universality
  13. Vertex-minor Ramsey and concluding remarks
  14. Acknowledgments.

Key Result

Theorem 1.1

For all $c>0$, if $G\sim \mathbb{G}(n,1/2)$ with $n\geq (1+c)\frac{1}{2\log_2(4/3)} k^2$, then $G$ is $k$-vertex-minor universal with probability at least $1-2^{-(1+o(1))ck^2/2}$.

Theorems & Definitions (90)

  • Definition
  • Definition
  • Theorem 1.1
  • Definition
  • Theorem 1.2
  • Theorem 1.3
  • Definition
  • Theorem 1.4
  • Conjecture 1.4
  • Lemma 2.1: Oum05
  • ...and 80 more