Nowhere-zero flow

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

${\displaystyle \sum _{e\in \delta ^{+}(v)}\varphi (e)=\sum _{e\in \delta ^{-}(v)}\varphi (e),}$

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.[1]

• The set of M-flows does not necessarily form a group as the sum of two flows on one edge may add to 0.
• (Tutte 1950) A graph G has an M-flow iff it has a |M|-flow. As a consequence, a ${\displaystyle \mathbb {Z} _{k}}$ flow exists iff a k-flow exists.[1] As a consequence G admits a k-flow then it admits an h-flow where ${\displaystyle h\geq k}$.
• Orientation independence. Modify a nowhere-zero flow φ on a graph G by choosing an edge e, reversing it, and then replacing φ(e) with −φ(e). After this adjustment, φ is still a nowhere-zero flow. Furthermore, if φ was originally a k-flow, then the resulting φ is also a k-flow. Thus, the existence of a nowhere-zero M-flow or a nowhere-zero k-flow is independent of the orientation of the graph. Thus, an undirected graph G is said to have a nowhere-zero M-flow or nowhere-zero k-flow if some (and thus every) orientation of G has such a flow.

Let ${\displaystyle N_{M}(G)}$ be the number of M-flows on G. It satisfies the deletion–contraction formula N(G) = N(G / e) − N(G \ e).[1]

Using this and induction, it can be shown that N(G) is a polynomial in ${\displaystyle |M|-1}$ where |M| is the order of the group M. We call N(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 ${\displaystyle N_{M_{1}}(G)=N_{M_{2}}(G)}$ if ${\displaystyle |M_{1}|=|M_{2}|}$.

The above results were proved by Tutte in 1953 when he was studying the Tutte polynomial, a generalization of the flow polynomial.[2]

There is a duality between region-colorings and M-flows, as well as the duality between k-region colorings and k-flows / Zk flows.

Let G be a directed bridgeless planar graph, and assume that the regions of this drawing are properly k-colored with the colors {0, 1, 2, .., k – 1}.

Construct a map φ: E(G) → {–(k – 1), ..., –1, 0, 1, ..., k – 1} by the following rule: if the edge e has a region of color x to the left and a region 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. Tutte proved that the converse is also true (use induction on |E(G)| ). This can be expressed concisely by the relation: [1]

${\displaystyle \chi (G^{*})=\phi (G)}$

where the RHS is the flow number, the smallest k for which G permits a k-flow.

• G is 2-region-colorable iff every vertex has even degree (consider NZ Z₂ flows).[1]
• Let ${\displaystyle K=\mathbb {Z} _{2}\times \mathbb {Z} _{2}}$ be the Klein-4 group. Then a cubic graph has a K-flow iff it is three-edge-colorable. As a corollary a cubic graph that is 3-edge colorable is 4-face colorable.[1]
• A graph is 4-face colorable iff it permits a NZ Z₄ flow (see Four color theorem). The Petersen graph does not have a NZ Z₄ flow, and this led to the 4-flow conjecture (see below).
• If G is a triangulation then G is 3-(vertex)colorable iff every vertex has even degree. By the first bullet, the dual graph G* is 2-colorable and thus bipartite and planar cubic. So G* has a NZ Z³ flow and is thus 3-region colorable, making G 3-vertex colorable.[1]
• Just as no graph with a loop edge has a proper (vertex) coloring, no graph with a bridge can have a NZ M-flow for any group M. Conversely, every bridgeless graph has a NZ Z-flow (a form of Robbins' theorem).[3]

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? (more unsolved problems in mathematics)

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[4]
• Seymour's 6-flow theorem: every bridgeless graph has a 6-flow (1981).[5]

• algebraic flow theory
• Cycle space
• Cycle double cover conjecture
• Four color theorem
• Graph coloring
• Edge coloring
• Tutte polynomial

• Zhang, Cun-Quan (1997). Integer Flows and Cycle Covers of Graphs. Chapman & Hall/CRC Pure and Applied Mathematics Series. Marcel Dekker, Inc. ISBN 9780824797904. LCCN 96037152.
• Zhang, Cun-Quan (2012). Circuit Double Cover of Graphs. Cambridge University Press. ISBN 978-0-5212-8235-2.
• Jensen, T. R.; Toft, B. (1995). "13 Orientations and Flows". Graph Coloring Problems. Wiley-Interscience Serires in Discrete Mathematics and Optimization. pp. 209–219.
• Jacobsen, Jesper Lykke; Salas, Jesús (2013). "Is the five-flow conjecture almost false?". Journal of Combinatorial Theory. Series B. 103 (4): 532–565. arXiv:1009.4062. doi:10.1016/j.jctb.2013.06.001. MR 3071381.