Concept in graph theory
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" 
formula_1
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:
formula_2
Flow polynomial.
Let formula_5 be the number of "M"-flows on "G". It satisfies the deletion–contraction formula:
formula_6
Combining this with induction we can show formula_5 is a polynomial in formula_8 where formula_9 is the order of the group "M". We call formula_5 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 formula_11 if formula_12
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 formula_13 Construct a map 
formula_14
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:
formula_15
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 formula_21 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.
Existence of "k"-flows.
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Unsolved problem in mathematics:
Does every bridgeless graph have a nowhere zero 5-flow? Does every bridgeless graph that does not have the Petersen graph as a minor have a nowhere zero 4-flow?
Interesting questions arise when trying to find nowhere-zero "k"-flows for small values of "k". The following have been proven:
Jaeger's 4-flow Theorem. Every 4-edge-connected graph has a 4-flow. 
Seymour's 6-flow Theorem. Every bridgeless graph has a 6-flow.
3-flow, 4-flow and 5-flow conjectures.
As of 2019, the following are currently unsolved (due to Tutte):
3-flow Conjecture. Every 4-edge-connected graph has a nowhere-zero 3-flow.
4-flow Conjecture. Every bridgeless graph that does not have the Petersen graph as a minor has a nowhere-zero 4-flow. 
5-flow Conjecture. Every bridgeless graph has a nowhere-zero 5-flow.
The converse of the 4-flow Conjecture does not hold since the complete graph "K"11 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.
References.
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