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.

Definition

Let G = (V,E) be a directed graph and let M be an abelian group. A map φ: E → M is a flow or an M-flow if for every vertex v ∈ V, it holds that

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

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. If M = Z is the group of integers under addition and k is a positive integer with the property that –k < φ(e) < k for every edge e, then the M-flow φ is also called a k-flow.

Let G = (V,E) be an undirected graph. An orientation of E is a modular k-flow if

$|\delta^+(v)| \equiv |\delta^-(v)| \pmod{k}$

for every vertex v ∈ V.

Properties

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.

More surprisingly, if M is a finite abelian group of size k, then the number of a nowhere-zero M-flows in some graph does not depend on the structure of M but only on k, the size of M. Furthermore, the existence of a M-flow coincides with the existence of a k-flow. These two results were proved by Tutte in 1953.^[1]

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 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. 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 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)

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 Z-flow (a form of Robbins theorem), but interesting questions arise when trying 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)^[2] and Seymour's 6-flow theorem (every bridgeless graph has a nowhere-zero 6-flow).^[3]

Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow^[4] and that every bridgeless graph that does not have the Petersen graph as a minor has a nowhere-zero 4-flow.^[5] 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.

See also

• Cycle space

References

1. Tutte, William Thomas (1953). "A contribution to the theory of chromatic polynomials".
2. F. Jaeger, Flows and generalized coloring theorems in graphs, J. Comb. Theory Set. B, 26 (1979), 205-216.
3. P. D. Seymour, Nowhere-zero 6-flows, J. Comb. Theory Ser B, 30 (1981), 130-135.
4. 5-flow conjecture, Open Problem Garden.
5. 4-flow conjecture, Open Problem Garden.

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