A nowhere-zero 
-flow
on a graph 
consists of a graph orientation
of its edges and an integer
value 
assigned to each graph edge such that 
and the sum of the values on edges
directed into each graph vertex equals the sum on
edges directed out of it (Tutte 1954). Existence of
such a flow is independent of the chosen graph orientation.
For a finite Abelian group of group order 
, a group-valued flow satisfies the same conservation rule
with addition in that group and assigns a nonzero value
to each graph edge. The number of such nowhere-zero
group-valued flows on a fixed graph orientation
is given by the flow polynomial evaluated at 
(Tutte 1954). This is not the count of
bounded integer-valued flows in the preceding definition.
For a bidirected graph, conservation uses the independent graph orientation at each end of a graph edge. A bidirected graph is flow-admissible if it has a nowhere-zero flow over the integers. Every bidirected graph that is flow-admissible and Eulerian has a nowhere-zero 4-flow. Chen and Fan (2026) strengthen this by showing that the edges on which the flow has magnitude 3 can all be confined to a single circle that is either unbalanced or has at least three edges.
For ordinary graph flows, a cycle change adds a fixed group element along a directed traversal of a graph cycle, subtracting it on edges directed against the traversal. Flow reconfiguration asks whether two nowhere-zero flows can be joined by finitely many such changes while all intermediate flows remain nowhere zero (Esperet et al. 2026).
Goel (2026, Theorem 1.1) proves that this reconfiguration space is connected for every finite multigraph
whose vertex degrees are even, for every finite Abelian group of group order
at least four. Loops contribute twice to vertex
degree. The analogous assertion holds for integer 
-flows for every 
, keeping every intermediate value in 
. Goel (2026) reports an AI-developed Lean
proof from the standard axioms.
