science

Kelmans–Seymour conjecture

Kelmans–Seymour 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 Kelmans–Seymour conjecture rather than just read about it. In short: In graph theory, the Kelmans–Seymour conjecture states that every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K5. It is named for Paul Seymour and Alexander Kelmans, who independently described the conjecture; Seymour in 1977 and Kelmans in 1979.

Key takeaways

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

Reference excerpt

In graph theory, the Kelmans–Seymour conjecture states that every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K5. It is named for Paul Seymour and Alexander Kelmans, who independently described the conjecture; Seymour in 1977 and Kelmans in 1979. A proof was announced in 2016, and published in four papers in 2020.

Formulation A graph is 5-vertex-connected when there are no five vertices that removed leave a disconnected graph. The complete graph is a graph with an edge between every five vertices, and a subdivision of a complete graph modifies this by replacing some of its edges by longer paths. So a graph G contains a subdivision of K5 if it is possible to pick out five vertices of G, and a set of ten paths connecting these five vertices in pairs without any of the paths sharing vertices or edges with each other. In any drawing of the graph on the Euclidean plane, at least two of the ten paths must cross each other, so a graph G that contains a K5 subdivision cannot be a planar graph. In the other direction, by Kuratowski's theorem, a graph that is not planar necessarily contains a subdivision of either K5 or of the complete bipartite graph K3,3. The Kelmans–Seymour conjecture refines this theorem by providing a condition under which only one of these two subdivisions, the subdivision of K5, can be guaranteed to exist. It states that, if a non-planar graph is 5-vertex-connected, then it contains a subdivision of K5.

Related results A related result, Wagner's theorem, states that every 4-vertex-connected nonplanar graph contains a copy of K5 as a graph minor. One way of restating this result is that, in these graphs, it is always possible to perform a sequence of edge contraction operations so that the resulting graph contains a K5 subdivision. The Kelmans–Seymour conjecture states that, with a higher order of connectivity, these contractions are not required. An earlier conjecture of Gabriel Andrew Dirac (1964), proven in 2001 by Wolfgang Mader, states that every n-vertex graph with at least 3n − 5 edges contains a subdivision of K5. Because planar graphs have at most 3n − 6 edges, the graphs with at least 3n − 5 edges must be nonplanar. However, they need not be 5-connected, and 5-connected graphs can have as few as 2.5n edges.

Claimed proof In 2016, a proof of the Kelmans–Seymour conjecture was claimed by Xingxing Yu of the Georgia Institute of Technology and his Ph.D. students Dawei He and Yan Wang. A sequence four papers proving this conjecture appeared in Journal of Combinatorial Theory, Series B.

See also Four-color theorem Hajós' conjecture

References

Illustrations

Worked examples

Example 1 — a first encounter with Kelmans–Seymour conjecture

Start with the simplest possible case. Write down what Kelmans–Seymour 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 Kelmans–Seymour 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 Kelmans–Seymour 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 Kelmans–Seymour conjecture

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

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

Frequently asked questions

What is Kelmans–Seymour conjecture in simple terms?

Why does Kelmans–Seymour 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 Kelmans–Seymour 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 Kelmans–Seymour conjecture.