Measurable Regular Subgraphs
Matt Bowen, Clinton T. Conley, Felix Weilacher

TL;DR
This paper characterizes when certain regular subgraphs in bipartite Borel graphs have Baire measurable spanning subgraphs, revealing a unique property related to parity and providing new insights into measurable graph colorings.
Contribution
It establishes a precise parity-based criterion for the existence of Baire measurable regular subgraphs in bipartite Borel graphs, including measure-theoretic analogs.
Findings
Baire measurable $k$-regular spanning subgraphs exist iff $d$ is odd or $k$ is even.
First example of a locally checkable coloring problem with measurable but not computable solutions.
Results extend to hyperfinite graphs in the measure setting.
Abstract
We show that every -regular bipartite Borel graph admits a Baire measurable -regular spanning subgraph if and only if is odd or is even. This gives the first example of a locally checkable coloring problem which is known to have a Baire measurable solution on Borel graphs but not a computable solution on highly computable graphs. We also prove the analogous result in the measure setting for hyperfinite graphs.
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Taxonomy
Topicsgraph theory and CDMA systems · Advanced Graph Theory Research · Rings, Modules, and Algebras
