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 $k\ge 4$, 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.