mathematics

Veblen's theorem

Veblen'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 Veblen's theorem rather than just read about it. In short: In mathematics, Veblen's theorem, introduced by Oswald Veblen (1912), states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. Thus, it is closely related to the theorem of Euler (1736) that a finite graph has an Euler tour (a single non-simple cycle that covers the edges of the graph) if and only if it is connected and every vert…

Key takeaways

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

Reference excerpt

In mathematics, Veblen's theorem, introduced by Oswald Veblen (1912), states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. Thus, it is closely related to the theorem of Euler (1736) that a finite graph has an Euler tour (a single non-simple cycle that covers the edges of the graph) if and only if it is connected and every vertex has even degree. Indeed, a representation of a graph as a union of simple cycles may be obtained from an Euler tour by repeatedly splitting the tour into smaller cycles whenever there is a repeated vertex. However, Veblen's theorem applies also to disconnected graphs, and can be generalized to infinite graphs in which every vertex has finite degree. If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by including more edges and vertices from G) to a finite Eulerian graph. In particular, every countably infinite graph with only one end and with no odd vertices can be written as a union of disjoint cycles.

See also Cycle basis Cycle double cover conjecture Eulerian matroid

Notes

References Euler, L. (1736), "Solutio problematis ad geometriam situs pertinentis" (PDF), Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8: 128–140. Reprinted and translated in Biggs, N. L.; Lloyd, E. K.; Wilson, R. J. (1976), Graph Theory 1736–1936, Oxford University Press. Sabidussi, Gert (1964), "Infinite Euler graphs", Canadian Journal of Mathematics, 16: 821–838, doi:10.4153/CJM-1964-078-x, MR 0169236. Veblen, Oswald (1912), "An Application of Modular Equations in Analysis Situs", Annals of Mathematics, Second Series, 14 (1): 86–94, doi:10.2307/1967604, JSTOR 1967604

Worked examples

Example 1 — a first encounter with Veblen's theorem

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

In research: Veblen'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 Veblen'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: Veblen'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 Veblen'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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Veblen's theorem in 20 minutes

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

Frequently asked questions

What is Veblen's theorem in simple terms?

Why does Veblen'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 Veblen'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 Veblen's theorem.