Nowhere-zero flow

In graph theory, nowhere-zero flows are a special type of network flow which is related (by duality) to coloring planar graphs. Let G = (V,E) be a directed graph and let M be an abelian group. We say that a map $\phi : E \rightarrow M$ is a flow or an M-flow if the following condition (sometimes called the Kirchoff Rule) is satisfied at every vertex $v \in V$ (here we let δ ⁺ (v) denote the set of edges pointing away from v and δ ⁻ (v) the set of edges pointing toward v).

$\sum_{e \in \delta^+(v)} \phi(e) = \sum_{e \in \delta^-(v)} \phi(e)$

If $\phi(e) \neq 0$ for every $e \in E$, we call ϕ a nowhere-zero flow. If $M = {\mathbb Z}$ and k is a positive integer with the property that − k < ϕ(e) < k for every edge e, we say that ϕ is a k-flow. Consider a flow ϕ on G and modify it by choosing an edge $e \in E$, reversing it, and then replacing ϕ(e) with − ϕ(e). After this adjustment, ϕ is still a flow, and further this adjustment preserves the properties k-flow and nowhere-zero flow. Thus, the existence of a nowhere-zero M-flow and the existence of a nowhere-zero k-flow is independent of the orientation of the graph, and we will say that an (undirected) graph has a nowhere-zero M-flow or k-flow if some (and thus every) orientation of it has such a flow.

Flow/coloring duality

Let G = (V,E) be a directed bridgeless graph drawn in the plane, and assume that the regions of this drawing are properly k-colored with the colors {0, 1, 2, .., k − 1}. Now, construct a map $\phi:E(G)\rightarrow\{-(k-1),\dots,-1,0,1,\dots,k-1\}$ 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. It is an easy exercise to show that ϕ is a k-flow. Furthermore, since the regions were properly colored, ϕ is a nowhere-zero k-flow. It follows from this construction, that if G and G * are planar dual graphs and G * is k-colorable, then G has a nowhere-zero k-flow. Tutte proved that the converse of this statement is also true. Thus, for planar graphs, nowhere-zero flows are dual to colorings. Since nowhere-zero flows make sense for general graphs (not just graphs drawn in the plane), this study can be viewed as an extension of coloring theory for non-planar graphs.

Theory

Unsolved problems 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?

Just as no graph with a loop edge has a proper coloring, no graph with a bridge can have a nowhere-zero flow (in any group). It is easy to show that every graph without a bridge has a nowhere-zero ${\mathbb Z}$-flow (a form of Robbins theorem), but interesting questions arise when we try to find nowhere-zero k-flows for small values of k. Two nice theorems in this direction are Jaeger's 4-flow theorem (every 4-edge-connected graph has a nowhere-zero 4-flow)^[1] and Seymour's 6-flow theorem (every bridgeless graph has a nowhere-zero 6-flow).^[2]

W. T. Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow^[3] and that every bridgeless graph that does not have the Petersen graph as a minor has a nowhere-zero 4-flow.^[4] For cubic graphs with no Petersen minor, a 4-flow is known to exist as a consequence of the snark theorem but for arbitrary graphs these conjectures remain open.

References

1. ^ F. Jaeger, Flows and generalized coloring theorems in graphs, J. Comb. Theory Set. B, 26 (1979), 205-216.
2. ^ P. D. Seymour, Nowhere-zero 6-flows, J. Comb. Theory Ser B, 30 (1981), 130-135.
3. ^ 5-flow conjecture, Open Problem Garden.
4. ^ 4-flow conjecture, Open Problem Garden.

• T.R. Jensen and B. Toft, Graph Coloring Problems, Wiley-Interscience Serires in Discrete Mathematics and Optimization, (1995)