Approximate Schreier decorations and approximate K\H{o}nig's line coloring Theorem

TL;DR

This paper extends recent finite graph results to Borel graphs, demonstrating that approximate Schreier decorations and Kőnig's line coloring can be achieved in the measurable setting for certain regular graphs.

Contribution

It proves the existence of approximate Schreier decorations and Kőnig's line coloring in Borel graphs, generalizing finite graph theorems to the measurable context.

Findings

Approximate Schreier decoration exists for 2Δ-regular Borel graphs.

Approximate Kőnig's line coloring holds for Borel graphs without odd cycles.

Approximate balanced orientation is established for even degree Borel graphs.

Abstract

Following recent result of L. M. T\' oth [arXiv:1906.03137] we show that every $2\Delta$-regular Borel graph $\mathcal{G}$ with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate K\H{o}nig's line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.