Graphings with few circulations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918256109944832 |
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| author | Kun, Gábor Tóth, László Márton |
| author_facet | Kun, Gábor Tóth, László Márton |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17071 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graphings with few circulations Kun, Gábor Tóth, László Márton Combinatorics Dynamical Systems 05C21, 37A15 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. |
| title | Graphings with few circulations |
| topic | Combinatorics Dynamical Systems 05C21, 37A15 |
| url | https://arxiv.org/abs/2512.17071 |