science

Woodall's conjecture

Woodall's conjecture is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Woodall's conjecture rather than just read about it. In short: In the mathematics of directed graphs, Woodall's conjecture is an unproven relationship between dicuts and dijoins. It was posed by Douglas Woodall in 1976.

Key takeaways

Woodall's conjecture belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Woodall's conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Woodall's conjecture from memory before moving on to harder problems.

Reference excerpt

In the mathematics of directed graphs, Woodall's conjecture is an unproven relationship between dicuts and dijoins. It was posed by Douglas Woodall in 1976.

Statement A dicut, in a directed graph, is a set of edges defined from a partition of the vertices into two subsets such that all edges that cross the partition do so in the same direction. A dijoin is a subset of edges that, when contracted, produces a strongly connected graph; equivalently, it is a subset of edges that includes at least one edge from each dicut.

According to the Lucchesi-Younger theorem, if the minimum number of edges in a dicut is k {\displaystyle k} , then there can be at most k {\displaystyle k} disjoint dijoins in the graph, because each one must include a different edge from the smallest dicut. Woodall's conjecture states that, in this case, it is always possible to find k {\displaystyle k} disjoint dijoins. That is, any directed graph the minimum number of edges in a dicut equals the maximum number of disjoint dijoins that can be found in the graph (a packing of dijoins).

Partial results It is a folklore result that the theorem is true for directed graphs whose minimum dicut has two edges. Any instance of the problem can be reduced to a directed acyclic graph by taking the condensation of the instance, a graph formed by contracting each strongly connected component to a single vertex. Another class of graphs for which the theorem has been proven true are the directed acyclic graphs in which every source vertex (a vertex without incoming edges) has a path to every sink vertex (a vertex without outgoing edges). In every graph whose minimum dicut size is k {\displaystyle k} , there exist at least ⌊ k / 6 ⌋ {\displaystyle \lfloor k/6\rfloor } disjoint dijoins. If a conjecture of W. T. Tutte on the existence of nowhere-zero 5-flows in biconnected undirected graphs is true, this bound would improve to ⌊ k / 5 ⌋ {\displaystyle \lfloor k/5\rfloor } .

Related results A fractional weighted version of the conjecture, posed by Jack Edmonds and Rick Giles, was refuted by Alexander Schrijver. In the other direction, the Lucchesi–Younger theorem states that the minimum size of a dijoin equals the maximum number of disjoint dicuts that can be found in a given graph.

References

External links Feofiloff, Paulo (November 30, 2005), Woodall's conjecture on Packing Dijoins: a survey (PDF) "Woodall's conjecture", Open Problem Garden, April 5, 2007

Worked examples

Example 1 — a first encounter with Woodall's conjecture

Start with the simplest possible case. Write down what Woodall's conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Woodall's conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Woodall's conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Woodall's conjecture

In research: Woodall's conjecture appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Woodall's conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Woodall's conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Directed graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Woodall's conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Woodall's conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Woodall's conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Woodall's conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Woodall's conjecture in simple terms?

In the mathematics of directed graphs, Woodall's conjecture is an unproven relationship between dicuts and dijoins. It was posed by Douglas Woodall in 1976.

Why does Woodall's conjecture matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Woodall's conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Woodall's conjecture.