It is intimately connected (by duality) to coloring planar graphs.
A map φ: E → M is an M-circulation if for every vertex v ∈ V where δ+(v) denotes the set of edges out of v and δ−(v) denotes the set of edges into v. Sometimes, this condition is referred to as Kirchhoff's law.
be the number of M-flows on G. It satisfies the deletion–contraction formula:[1] Combining this with induction we can show
the flow polynomial of G and abelian group M. The above implies that two groups of equal order have an equal number of NZ flows.
The order is the only group parameter that matters, not the structure of M. In particular
[2] There is a duality between k-face colorings and k-flows for bridgeless planar graphs.
To see this, let G be a directed bridgeless planar graph with a proper k-face-coloring with colors
So if G and G* are planar dual graphs and G* is k-colorable (there is a coloring of the faces of G), then G has a NZ k-flow.
This can be expressed concisely as:[1] where the RHS is the flow number, the smallest k for which G permits a k-flow.
Given this duality between NZ flows and colorings, and since we can define NZ flows for arbitrary graphs (not just planar), we can use this to extend face-colorings to non-planar graphs.
[1] Interesting questions arise when trying to find nowhere-zero k-flows for small values of k. The following have been proven: As of 2019, the following are currently unsolved (due to Tutte): The converse of the 4-flow Conjecture does not hold since the complete graph K11 contains a Petersen graph and a 4-flow.
[1] For bridgeless cubic graphs with no Petersen minor, 4-flows exist by the snark theorem (Seymour, et al 1998, not yet published).
The four color theorem is equivalent to the statement that no snark is planar.