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The Cranston--Li--Su--Wang--Xu (2026) conjecture is true — arXiv:2610.40863

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.40863
License (as archived): CC0-1.0
Archived: 2026-11-02T08:00:01.933Z
SHA-256 of the archived text: 2eebe8b66f982887cd1c665d37f66b737ff7f32e993ba53158038701f3ecf43e
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How to cite

The Cranston--Li--Su--Wang--Xu (2026) conjecture is true — arXiv:2610.40863 — arXiv (en), https://arxiv.org/abs/2610.40863 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bf586bdb87ec (Metraipolis archive dc_bf586bdb87ec, sha256 2eebe8b66f982887cd1c665d37f66b737ff7f32e993ba53158038701f3ecf43e)

BibTeX

@misc{dcbf586bdb87ec,
  title = {The Cranston--Li--Su--Wang--Xu (2026) conjecture is true — arXiv:2610.40863},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.40863},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bf586bdb87ec; sha256 2eebe8b66f982887cd1c665d37f66b737ff7f32e993ba53158038701f3ecf43e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.40863. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $A$ be a finite abelian group. A nowhere-zero $A$-flow of a graph gives every edge a direction and a non-zero value in $A$, so that at every vertex the sum of the values on the entering edges equals the sum on the leaving edges. Such a flow 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 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 $k\ge 4$, 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 core of the proof is an induction on the number of edges: if a flow enters a vertex $v$ along an edge $uv$ and leaves $v$ along an edge $vw$ with the same value, the two edges can be replaced by one edge $uw$, and the induction hypothesis applies to the smaller graph. So the main task is to change every flow into a flow with such a pair of edges. For nowhere-zero $A$-flows with $|A|\ge 4$, replacing vertices of degree more than $4$ by triangles reduces the problem to graphs with all degrees at most $4$. For integer flows, a counting argument joins any two nowhere-zero flows with values in the cyclic group $\mathbb{Z}_k$, $k\ge 4$, by changes that add $1$, $-1$, $2$ or $-2$; every such chain gives a chain of integer flows with values between $1$ and $k-1$ in absolute value, and every integer flow of this chain reduces modulo $k$ to a flow of the given chain. All results of this paper are also proved in the Lean 4 proof assistant, from Lean's standard axioms alone.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.40863 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-02T07:42:45.724Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).