In graph theory, a nowhere-zero flow or NZ flow is a network flow that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs.

Definitions

Let G = (V,E) be a digraph and let M be an abelian group. 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. If φ(e) ≠ 0 for every e ∈ E, we call φ a nowhere-zero flow, an M-flow,** or an **NZ-flow. If k is an integer and 0 < |φ(e)| < k then φ is a k-flow.

Other notions

Let G = (V,E) be an undirected graph. An orientation of E is a modular k-flow if for every vertex v ∈ V we have:

Properties

Flow polynomial

Let N_M(G) be the number of M-flows on G. It satisfies the deletion–contraction formula: Combining this with induction we can show N_M(G) is a polynomial in |M|-1 where |M| is the order of the group M. We call N_M(G) 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 if The above results were proved by Tutte in 1953 when he was studying the Tutte polynomial, a generalization of the flow polynomial.

Flow-coloring duality

Bridgeless Planar Graphs

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 Construct a map by the following rule: if the edge e has a face of color x to the left and a face of color y to the right, then let φ(e) = x – y. Then φ is a (NZ) k-flow since x and y must be different 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. Using induction on |E(G)| Tutte proved the converse is also true. This can be expressed concisely as: where the RHS is the flow number, the smallest k for which G permits a k-flow.

General Graphs

The duality is true for general M-flows as well: The duality follows by combining the last two points. We can specialize to M = \Z_k to obtain the similar results for k-flows discussed above. 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.

Applications

Existence of k-flows

Interesting questions arise when trying to find nowhere-zero k-flows for small values of k. The following have been proven:

3-flow, 4-flow and 5-flow conjectures

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. 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.