ArticleslgStudy

science

Poussin graph

Poussin graph 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 Poussin graph rather than just read about it. In short: In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin.

Poussin graph — main illustration
Poussin graph — illustration

Key takeaways

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

Reference excerpt

In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin.

History In 1879, Alfred Kempe published a proof of the four color theorem, one of the big conjectures in graph theory. While the theorem is true, Kempe's proof is incorrect. Percy John Heawood illustrated it in 1890 with a counter-example, and de la Vallée-Poussin reached the same conclusion in 1896 with the Poussin graph. Kempe's (incorrect) proof is based on alternating chains, and as those chains prove useful in graph theory mathematicians remain interested in such counterexamples. More were found later: first, the Errera graph in 1921, then the Kittell graph in 1935, with 23 vertices, and finally two minimal counter-examples (the Soifer graph in 1997 and the Fritsch graph in 1998, both of order 9).

References

External links Eric W. Weisstein, Poussin Graph (MathWorld)

Illustrations

Poussin graph illustration
Poussin graph: Tangled Kempe chains in the Poussin graph. The adjacencies between regions of this map form the Poussin graph, partially four-colored with the outer region uncolored. The blue–yellow and blue–green Kempe chains (yellow and green lines) connect the outer region's neighbors, so Kempe would swap colors in the left red–yellow chain and the right red–green chain (red lines), allowing the outer region to be red. As the blue–yellow and blue–green chains cross, this color swap would cause the upper yellow and green regions to both become red, producing an invalid coloring.
Tangled Kempe chains in the Poussin graph. The adjacencies between regions of this map form the Poussin graph, partially four-colored with the outer region uncolored. The blue–yellow and blue–green Kempe chains (yellow and green lines) connect the outer region's neighbors, so Kempe would swap colors in the left red–yellow chain and the right red–green chain (red lines), allowing the outer region to be red. As the blue–yellow and blue–green chains cross, this color swap would cause the upper yellow and green regions to both become red, producing an invalid coloring.

Worked examples

Example 1 — a first encounter with Poussin graph

Start with the simplest possible case. Write down what Poussin graph 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 Poussin graph 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 Poussin graph 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 Poussin graph

In research
Poussin graph 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 Poussin graph 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
Poussin graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Individual graphs, so understanding it makes those chapters shorter.
In everyday life
Look for Poussin graph 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 Poussin graph in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Poussin graph means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Poussin graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Poussin graph in simple terms?

In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin.

Why does Poussin graph 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 Poussin graph?

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 Poussin graph.

Tags

  • Individual graphs

Keep exploring