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Graphings with few circulations

Gábor Kun, László Márton Tóth

TL;DR

The authors develop a flexible inverse-limit construction to produce d-regular graphings (treeings) with precisely k-dimensional spaces of circulations in the measurable setting, for any d≥3 and any finite k. The core idea is a diagonalization that lifts a finite set of prescribed circulations while eliminating all others via carefully controlled surgeries and boundary-defect management, ensuring convergence to a treeing with the desired circulation space. They then show how this framework yields three concrete applications: a treeing with a single balanced orientation but no Schreier decoration, a treeing that is a Schreier graphing for 𝔽_d with no extra circulations, and a treeing with a proper d-edge-coloring but no additional perfect matchings. The results answer longstanding questions about decompositions of circulations into finite cycles and infinite paths and open new directions in the study of measurable combinatorics, group actions, and Markov-space generalizations. This work thus provides both a robust construction technique and a suite of structural obstructions that sharpen our understanding of circulations in graphings.

Abstract

In 2021, motivated by graph limit theory Lovász 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 Lovász, as this particular balanced orientation does not decompose as a linear combination of finite cycles and infinite paths.

Graphings with few circulations

TL;DR

The authors develop a flexible inverse-limit construction to produce d-regular graphings (treeings) with precisely k-dimensional spaces of circulations in the measurable setting, for any d≥3 and any finite k. The core idea is a diagonalization that lifts a finite set of prescribed circulations while eliminating all others via carefully controlled surgeries and boundary-defect management, ensuring convergence to a treeing with the desired circulation space. They then show how this framework yields three concrete applications: a treeing with a single balanced orientation but no Schreier decoration, a treeing that is a Schreier graphing for 𝔽_d with no extra circulations, and a treeing with a proper d-edge-coloring but no additional perfect matchings. The results answer longstanding questions about decompositions of circulations into finite cycles and infinite paths and open new directions in the study of measurable combinatorics, group actions, and Markov-space generalizations. This work thus provides both a robust construction technique and a suite of structural obstructions that sharpen our understanding of circulations in graphings.

Abstract

In 2021, motivated by graph limit theory Lovász extended most of the theory of flows to a measure theoretic setting. Using this framework, the first author constructed -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 -regular treeings where the space of circulations is exactly -dimensional for any positive integer . 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 -coloring, but no further perfect matchings. The first answers a question raised by Lovász, as this particular balanced orientation does not decompose as a linear combination of finite cycles and infinite paths.

Paper Structure

This paper contains 23 sections, 12 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Acknowledgements.
  4. Preliminaries
  5. Graphings and circulations
  6. Inverse limits
  7. Finite number of circulations
  8. Strategy
  9. Glossary
  10. Construction
  11. General setup
  12. Deficiency near the boundary
  13. Surgery and extension.
  14. $\mathcal{T}$ as an inverse limit
  15. The elimination of the other circulations
  16. ...and 8 more sections

Key Result

Theorem 1.1

For every degree $d \geq 3$ and finite number $k \in \mathbb{N}$ there exists a $d$-regular treeing $\mathcal{T}$ such that the vector space of circulations in $L^2(E(\mathcal{T}), \eta )$ is exactly $k$ dimensional.

Figures (1)

  • Figure 1: Outline of the proof of Theorem \ref{['thm:finitely_many_circ']}

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 21 more