Graphings with few circulations

TL;DR

This paper develops methods to construct regular graphings with a prescribed finite-dimensional space of circulations, leading to new examples that challenge existing assumptions about graphings and flows.

Contribution

It introduces tools to build regular graphings with exactly k-dimensional circulation spaces, providing new counterexamples and answering a question by Lovász.

Findings

Constructed graphings with exactly one circulation

Created graphings with specific orientations and colorings

Provided counterexamples to previous assumptions

Abstract

In 2021, motivated by graph limit theory Lov\'asz extended most of the theory of flows to a measure theoretic setting. Using this framework, the first author constructed $d$-regular treeings that are measurably bipartite, and have no nonzero measurable circulations, that is, flows without sources or sinks. In particular, these treeings do not admit a measurable perfect matching. In this paper, we develop tools to build $d$-regular treeings where the space of circulations is exactly $k$-dimensional for any positive integer $k$. As applications, we construct 1) a treeing with a single balanced orientation, but no Schreier decoration; 2) a treeing with a single Schreier decoration; 3) and a treeing with a proper edge $d$-coloring, but no further perfect matchings. The first answers a question raised by Lov\'asz, as this particular balanced orientation does not decompose as a linear combination of finite cycles and infinite paths.