Measurable Regular Subgraphs

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 $d$-regular bipartite Borel graph admits a Baire measurable $k$-regular spanning subgraph if and only if $d$ is odd or $k$ 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.