One-ended spanning trees and definable combinatorics

TL;DR

This paper proves the existence of Borel one-ended spanning trees in certain graphs on Polish spaces, extending previous measure-preserving results to more general settings and applications.

Contribution

It generalizes the existence of Borel one-ended spanning trees to non-measure-preserving contexts and applies this to orientations and perfect matchings in graphs.

Findings

Borel one-ended spanning trees exist generically in certain graphs.

Results extend to graphs induced by amenable groups and polynomial growth groups.

Baire measurable perfect matchings exist in bipartite regular graphs.

Abstract

Let $(X,\tau)$ be a Polish space with Borel probability measure $\mu,$ and $G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a Borel one-ended spanning tree generically. If $G$ is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result $\mu$-a.e. (resp., everywhere). Our results generalize recent work of Tim\'ar, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even degree graphs and measurable and Baire measurable perfect matchings in regular bipartite graphs, refining theorems that were previously only known to hold for measure preserving graphs. In particular, we prove that bipartite one-ended $d$-regular Borel graphs admit Baire measurable perfect matchings.