mathematics

Pósa's theorem

Pósa's 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 Pósa's theorem rather than just read about it. In short: Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem.

Key takeaways

Pósa's 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 Pósa's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pósa's theorem from memory before moving on to harder problems.

Reference excerpt

Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem. Unlike those conditions, it can be applied to graphs with a small number of low-degree vertices. It is named after Lajos Pósa, a protégé of Paul Erdős born in 1947, who discovered this theorem in 1962. The Pósa condition for a finite undirected graph G {\displaystyle G} having n {\displaystyle n} vertices requires that, if the degrees of the n {\displaystyle n} vertices in increasing order as

d 1 ≤ d 2 ≤ . . . ≤ d n , {\displaystyle d_{1}\leq d_{2}\leq ...\leq d_{n},}

then for each index k < n / 2 {\displaystyle k<n/2} the inequality k < d k {\displaystyle k<d_{k}} is satisfied. Pósa's theorem states that if a finite undirected graph satisfies the Pósa condition, then that graph has a Hamiltonian cycle in it.

References Pósa, L. (1962), "A theorem concerning Hamilton lines", Magyar Tud. Akad. Mat. Kutató Int. Közl., 7: 225–226, MR 0184876 Katona–Recski–Szabó: A számítástudomány alapjai, Typotex, Budapest, 2003, (Hungarian undergraduate level course book). Kronk, Hudson V. (1969), "Generalization of a theorem of Pósa", Proceedings of the American Mathematical Society, 21 (1): 77–78, doi:10.2307/2036861, JSTOR 2036861, MR 0237377 Kühn, Daniela; Osthus, Deryk; Treglown, Andrew (2009), "Degree sequences forcing Hamilton cycles in directed graphs", European Conference on Combinatorics, Graph Theory and Applications (EuroComb 2009), Electron. Notes Discrete Math., vol. 34, Amsterdam: Elsevier Sci. B. V., pp. 347–351, doi:10.1016/j.endm.2009.07.057, MR 2591466 Yin, Jian-Hua; Zhang, Yue (2011), "Pósa-condition and nowhere-zero 3-flows", Discrete Mathematics, 311 (12): 897–907, doi:10.1016/j.disc.2011.02.023, MR 2787300

External links Weisstein, Eric W., "Pósa's Theorem", MathWorld About the Pósa theorem

Worked examples

Example 1 — a first encounter with Pósa's theorem

Start with the simplest possible case. Write down what Pósa's 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 Pósa's 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 Pósa's 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 Pósa's theorem

In research: Pósa's 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 Pósa's 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: Pósa's 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 Pósa's 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 Pósa's theorem in 20 minutes

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

Frequently asked questions

What is Pósa's theorem in simple terms?

Why does Pósa's 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 Pósa's 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 Pósa's theorem.