mathematics

Hanani–Tutte theorem

Hanani–Tutte 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 Hanani–Tutte theorem rather than just read about it. In short: In topological graph theory, the Hanani–Tutte theorem is a result on the parity of edge crossings in a graph drawing. It states that every drawing in the plane of a non-planar graph contains a pair of independent edges (not both sharing an endpoint) that cross each other an odd number of times.

Key takeaways

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

Reference excerpt

In topological graph theory, the Hanani–Tutte theorem is a result on the parity of edge crossings in a graph drawing. It states that every drawing in the plane of a non-planar graph contains a pair of independent edges (not both sharing an endpoint) that cross each other an odd number of times. Equivalently, it can be phrased as a planarity criterion: a graph is planar if and only if it has a drawing in which every pair of independent edges crosses evenly (or not at all).

History The result is named after Haim Hanani, who proved in 1934 that every drawing of the two minimal non-planar graphs K5 and K3,3 has a pair of edges with an odd number of crossings, and after W. T. Tutte, who stated the full theorem explicitly in 1970. A parallel development of similar ideas in algebraic topology has been credited to Egbert van Kampen, Arnold S. Shapiro, and Wu Wenjun.

Applications One consequence of the theorem is that testing whether a graph is planar may be formulated as solving a system of linear equations over the finite field of order two. These equations may be solved in polynomial time, but the resulting algorithms are less efficient than other known planarity tests.

Generalizations For other surfaces S than the plane, a graph can be drawn on S without crossings if and only if it can be drawn in such a way that all pairs of edges cross an even number of times; this is known as the weak Hanani–Tutte theorem for S. The strong Hanani–Tutte theorem states that a graph can be drawn without crossings on S if and only if it can be drawn in such a way that all independent pairs of edges cross an even number of times, without regard for the number of crossings between edges that share an endpoint; this strong version does not hold for all surfaces, but it is known to hold for the plane, the projective plane and the torus. The same approach, in which one shows that pairs of edges with an even number of crossings can be disregarded or eliminated in some type of graph drawing and uses this fact to set up a system of linear equations describing the existence of a drawing, has been applied to several other graph drawing problems, including upward planar drawings, drawings minimizing the number of uncrossed edges, and clustered planarity.

References

Worked examples

Example 1 — a first encounter with Hanani–Tutte theorem

Start with the simplest possible case. Write down what Hanani–Tutte 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 Hanani–Tutte 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 Hanani–Tutte 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 Hanani–Tutte theorem

In research: Hanani–Tutte 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 Hanani–Tutte 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: Hanani–Tutte theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph drawing, Statements about planar graphs, Theorems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Hanani–Tutte 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 Hanani–Tutte theorem in 20 minutes

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

Frequently asked questions

What is Hanani–Tutte theorem in simple terms?

Why does Hanani–Tutte 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 Hanani–Tutte 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 Hanani–Tutte theorem.