Finitely dependent random colorings of bounded degree graphs

TL;DR

This paper constructs finitely dependent proper colorings for graphs with bounded degree, including infinite graphs like ^d and regular trees, using probabilistic and automorphism-invariant methods.

Contribution

It introduces new finitely dependent colorings for bounded degree graphs, including infinite cases, with explicit constructions as factors of iid and automorphism-invariant colorings.

Findings

Existence of 4-dependent proper colorings with a specific number of colors for graphs of degree d.

Construction of 2-dependent automorphism-invariant colorings for unimodular transitive graphs.

Application to ^d and regular trees, providing explicit colorings with finite dependence.

Abstract

We prove that every (possibly infinite) graph of degree at most $d$ has a 4-dependent random proper $4^{d(d+1)/2}$-coloring, and one can construct it as a finitary factor of iid. For unimodular transitive (or unimodular random) graphs we construct an automorphism-invariant (respectively, unimodular) 2-dependent coloring by $3^{d(d+1)/2}$ colors. In particular, there exist random proper colorings for $\Z^d$ and for the regular tree that are 2-dependent and automorphism-invariant, or 4-dependent and finitary factor of iid.