math.CO — Combinatorics
The Cranston–Li–Su–Wang–Xu (2026) conjecture is true
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.