Whitney's 2-isomorphism theorem for graphings
Abstract
We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. For locally finite graphings, we define a weak isomorphism to be an edge-measure-preserving Borel bijection that preserves cycles and hyperfinite subgraphs, modulo null sets. Our rigidity theorem shows that every weak isomorphism of a weakly $3$-connected, infinitely-ended graphing is induced by an isomorphism of graphings. To our knowledge, this is the first general sufficient condition in measurable combinatorics for two given graphings to be isomorphic. We then prove a full measurable Whitney theorem, namely that every weak isomorphism between graphings is implemented by a finite composition of locally finite sequences of six measurable Whitney operations. The proofs combine measurable-combinatorial techniques, applications of spanning forests from percolation theory, and graph decomposition arguments. A central ingredient is the construction and analysis of infinitely-ended leafless subforests. With these tools we also show that, between weakly $2$-connected, infinitely-ended graphings, preservation of hyperfiniteness alone forces preservation of cycles, which is of independent interest. This work further develops the limit theory of matroids recently initiated by Lovász and provides its first application in finite graph theory. We show that Whitney's rigidity theorem is robust under sublinear errors for sequences of finite graphs with uniformly bounded degrees and suitable connectivity.
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BibTeX
@misc{borbenyi2026whitneys,
title = {{Whitney's 2-isomorphism theorem for graphings}},
author = {Márton Borbényi and Grigory Terlov and László Márton Tóth},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.16152},
eprint = {2605.16152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.16152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Whitney's 2-isomorphism theorem for graphings
AU - Márton Borbényi
AU - Grigory Terlov
AU - László Márton Tóth
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.16152
UR - https://arxiv.org/abs/2605.16152
AB - We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. For locally finite graphings, we define a weak isomorphism to be an edge-measure-preserving Borel bijection that preserves cycles and hyperfinite subgraphs, modulo null sets. Our rigidity theorem shows that every weak isomorphism of a weakly $3$-connected, infinitely-ended graphing is induced by an isomorphism of graphings. To our knowledge, this is the first general sufficient condition in measurable combinatorics for two given graphings to be isomorphic. We then prove a full measurable Whitney theorem, namely that every weak isomorphism between graphings is implemented by a finite composition of locally finite sequences of six measurable Whitney operations. The proofs combine measurable-combinatorial techniques, applications of spanning forests from percolation theory, and graph decomposition arguments. A central ingredient is the construction and analysis of infinitely-ended leafless subforests. With these tools we also show that, between weakly $2$-connected, infinitely-ended graphings, preservation of hyperfiniteness alone forces preservation of cycles, which is of independent interest. This work further develops the limit theory of matroids recently initiated by Lovász and provides its first application in finite graph theory. We show that Whitney's rigidity theorem is robust under sublinear errors for sequences of finite graphs with uniformly bounded degrees and suitable connectivity.
ER -