mathematics

Fulkerson–Chen–Anstee theorem

Fulkerson–Chen–Anstee theorem is a mathematics 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 Fulkerson–Chen–Anstee theorem rather than just read about it. In short: The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers ( ( a 1 , b 1 ) , … , ( a n , b n ) ) {\displaystyle ((a_{1},b_{1}),\ldots ,(a_{n},b_{n}))} to be the indegree-outdegree pairs of a simple directed graph; a sequen…

Key takeaways

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

Reference excerpt

The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers ( ( a 1 , b 1 ) , … , ( a n , b n ) ) {\displaystyle ((a_{1},b_{1}),\ldots ,(a_{n},b_{n}))} to be the indegree-outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. Fulkerson (1960) obtained a characterization analogous to the classical Erdős–Gallai theorem for graphs, but in contrast to this solution with exponentially many inequalities. In 1966 Chen improved this result in demanding the additional constraint that the integer pairs must be sorted in non-increasing lexicographical order leading to n inequalities. Anstee (1982) observed in a different context that it is sufficient to have a 1 ≥ ⋯ ≥ a n {\displaystyle a_{1}\geq \cdots \geq a_{n}} . Berger reinvented this result and gives a direct proof.

Statement A sequence ( ( a 1 , b 1 ) , … , ( a n , b n ) ) {\displaystyle ((a_{1},b_{1}),\ldots ,(a_{n},b_{n}))} of nonnegative integer pairs with a 1 ≥ ⋯ ≥ a n {\displaystyle a_{1}\geq \cdots \geq a_{n}} is digraphic if and only if ∑ i = 1 n a i = ∑ i = 1 n b i {\displaystyle \sum _{i=1}^{n}a_{i}=\sum _{i=1}^{n}b_{i}} and the following inequality holds for k such that 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} :

∑ i = 1 k a i ≤ ∑ i = 1 k min ( b i , k − 1 ) + ∑ i = k + 1 n min ( b i , k ) {\displaystyle \sum _{i=1}^{k}a_{i}\leq \sum _{i=1}^{k}\min(b_{i},k-1)+\sum _{i=k+1}^{n}\min(b_{i},k)}

Stronger versions Berger proved that it suffices to consider the k {\displaystyle k} th inequality such that 1 ≤ k < n {\displaystyle 1\leq k<n} with a k > a k + 1 {\displaystyle a_{k}>a_{k+1}} and for k = n {\displaystyle k=n} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Fulkerson–Chen–Anstee theorem

Start with the simplest possible case. Write down what Fulkerson–Chen–Anstee theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Fulkerson–Chen–Anstee theorem 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 Fulkerson–Chen–Anstee theorem 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 Fulkerson–Chen–Anstee theorem

In research: Fulkerson–Chen–Anstee theorem appears in mathematics 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 Fulkerson–Chen–Anstee theorem 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: Fulkerson–Chen–Anstee theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Fulkerson–Chen–Anstee theorem 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 Fulkerson–Chen–Anstee theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Fulkerson–Chen–Anstee theorem 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 Fulkerson–Chen–Anstee theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Fulkerson–Chen–Anstee theorem in simple terms?

Why does Fulkerson–Chen–Anstee theorem matter?

Because it connects several mathematics 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 Fulkerson–Chen–Anstee theorem?

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 Fulkerson–Chen–Anstee theorem.