Invariant Gaussian processes and independent sets on regular graphs of large girth

TL;DR

This paper improves the lower bound on the size of independent sets in large-girth 3-regular graphs using invariant Gaussian processes and factor of i.i.d. constructions, advancing understanding of combinatorial properties of such graphs.

Contribution

It introduces a novel method employing invariant Gaussian processes on regular trees to approximate eigenvector equations, leading to improved bounds on independent set sizes.

Findings

Established a new lower bound of 0.4361n for independent sets in large-girth 3-regular graphs.

Showed that invariant Gaussian processes can be approximated by i.i.d. factors for certain eigenvalues.

Provided evidence via simulation that the bound could be around 0.438n.

Abstract

We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue \lambda. We show that such processes can be approximated by i.i.d. factors provided that $|\lambda| \leq 2\sqrt{d-1}$. We then use these approximations for $\lambda = -2\sqrt{d-1}$ to produce factor of i.i.d. independent sets on regular trees.