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The Cranston–Li–Su–Wang–Xu (2026) conjecture is true

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Dave Goel ↗

Version 1 / Nov 12, 2026/ arXiv: 2610.40863 / CC BY-SA 4.0

Abstract

Let A be a finite abelian group. A nowhere-zero A-flow of a graph can be changed along a cycle: going around the cycle, add a fixed non-zero element of A to the value of every edge that points forward and subtract it from the value of every edge that points backward. The Cranston–Li–Su–Wang–Xu (2026) conjecture on the reconfiguration of nowhere-zero flows states that if A has at least four elements, then in every graph in which all degrees are even, any two nowhere-zero A-flows are joined by a chain of such changes through nowhere-zero A-flows. It also states that, for every integer k ≥ 4, the same holds for integer flows whose values are between 1 and k − 1 in absolute value, where each change adds a non-zero integer. We prove this conjecture. The proof is by induction on the number of edges. If a flow enters a vertex along one edge and leaves it along another edge with the same value, the two edges can be replaced by a single edge; this gives a smaller graph. The main step is to change every flow into a flow with such a pair of edges. For A-flows we do this when all degrees are at most 4; repeatedly replacing a vertex of degree at least 6 by a triangle, following Cranston et al., reduces the general case to this one. For integer flows, we show that, in a graph in which all degrees are even, any two nowhere-zero flows with values in the cyclic group Z_k, k ≥ 4, are joined by changes that add only 1, −1, 2 or −2, and that such changes lift to integer flows. All theorems of this paper are also proved in the Lean 4 proof assistant.

Author comments: An expanded version of this paper is posted on Zenodo: https://doi.org/10.5281/zenodo.23605838

Provenance statement

This work was done overwhelmingly by AI models and supervised by the author. The models were Anthropic's Claude and OpenAI's ChatGPT. The author takes full responsibility for the paper.

References

  1. D. W. Cranston, J. Li, B. Su, Z. Wang, N. Xu, Reconfiguration of nowhere-zero flows , arXiv:2606.24685v1 (June 23, 2026). Statement numbers refer to this version.arXiv
  2. L. Esperet, K. Hendrey, A. Lagoutte, M. Marseloo, S. Norin, R. Steiner, Nowhere-zero flow reconfiguration , arXiv:2512.17342v4 (July 3, 2026). Statement numbers refer to this version.arXiv
  3. D. Goel, The Cranston--Li--Su--Wang--Xu (2026) conjecture is true, with extensions to graphs with few vertices of odd degree , Zenodo (2026), doi:10.5281/zenodo.23605838 (https://doi.org/10.5281/zenodo.23605838)DOI
  4. D. Goel, Flow reconfiguration with one modulus for all graphs, and Cereceda's conjecture for bipartite 22-degenerate graphs , Zenodo, version 2.30 (October 23, 2026). Statement numbers refer to this version. doi:10.5281/zenodo.23603238 (https://doi.org/10.5281/zenodo.23603238)DOI
  5. D. Goel, Nowhere-zero flow reconfiguration in every finite abelian group of order at least 30243024 , Zenodo, version 1.20 (October 23, 2026). Statement numbers refer to this version. doi:10.5281/zenodo.23605837 (https://doi.org/10.5281/zenodo.23605837)DOI
  6. F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs---a nonhomogeneous analogue of nowhere-zero flow properties , J. Combin. Theory Ser. B 56 (1992), no. 2, 165--182.
  7. W. T. Tutte, A contribution to the theory of chromatic polynomials , Canad. J. Math. 6 (1954), 80--91.

Version history

  1. v1 · 15 pagesSubmitted by Dave GoelAn expanded version of this paper is posted on Zenodo: https://doi.org/10.5281/zenodo.23605838CurrentNov 12, 2026