The Cranston--Li--Su--Wang--Xu (2026) conjecture is true
Abstract
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≥ 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|≥ 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 ℤk, k≥ 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.
Related
BibTeX
@misc{goel2026cranstonlisuwangxu,
title = {{The Cranston--Li--Su--Wang--Xu (2026) conjecture is true}},
author = {Dave Goel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40863},
eprint = {2610.40863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40863},
}
RIS
TY - UNPB
TI - The Cranston--Li--Su--Wang--Xu (2026) conjecture is true
AU - Dave Goel
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40863
UR - https://arxiv.org/abs/2610.40863
AB - 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≥ 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|≥ 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 ℤ_k, k≥ 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.
ER -