October 2, 2025Open Access

Measurable one-ended spanning trees

Key Points

A measurable one-ended spanning tree exists if the graph is measure-hyperfinite, showing a direct link between these concepts.
This study confirms a question posed by Bowen, Poulin, and Zomback regarding measurable graphs and their spanning properties.
It builds upon recent findings by Timár, Conley, Gaboriau, Marks, and Tucker-Drob, enriching the understanding of measurable structures.
The results may lead to further explorations in the area of measurable graph theory and its applications in probability.

Abstract

We show that a one-ended, locally finite, measurable graph on a standard probability space admits a measurable one-ended spanning subtree if and only if it is measure-hyperfinite. This answers a question posed by Bowen, Poulin, and Zomback and extends recent results of Timár and Conley, Gaboriau, Marks, and Tucker-Drob.

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Cite This Study

Bowen et al. (Thu,) studied this question.

synapsesocial.com/papers/68de5d9c83cbc991d0a205c0 https://doi.org/https://doi.org/10.48550/arxiv.2509.15000