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Combinatorics and Geometry

There are many problems in geometry that are innately combinatorial or for which combinatorial techniques shed light on the problem. ExampleIn , we show a family of 4 lines in the plane.

Combinatorics and Geometry

There are many problems in geometry that are innately combinatorial or for which combinatorial techniques shed light on the problem.

Example

In , we show a family of \(4\) lines in the plane. Each pair of lines intersects and no point in the plane belongs to more than two lines. These lines determine\(11\) regions.

Under these same restrictions, how many regions would a family of \(8947\) lines determine? Can different arrangements of lines determine different numbers of regions?

Example

Mandy says she has found a set of\(882\) points in the plane that determine exactly \(752\) lines. Tobias disputes her claim. Who is right?

Example

There are many different ways to draw a graph in the plane. Some drawings may have crossing edges while others don't. But sometimes, crossing edges must appear in any drawing. Consider the graph \(G\)

Can you redraw \(G\) without crossing edges?

Suppose Sam and Deborah were given a homework problem asking whether a particular graph on \(2843952\) vertices and \(9748032\) edges could be drawn without edge crossings. Deborah just looked at the number of vertices and the number of edges and said that the answer is no. Sam questions how she can be so certainwithout looking more closely at the structure of the graph. Is there a way for Deborah to justify her definitive response?

Tani ti. Asnjë kalkulator nuk e zgjidh këtë, por pjesët e saj janë të llogaritura. Provo një më poshtë, ose shkruaj tënde.

Mbaje punën tënde

Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.

Regjistrohu Hyr

Simbolet e përdorura këtu

Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.

Pyetja që bëjnë njerëzit

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Pjesa e kësaj faqeje është adaptuar nga Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). E përmbledhur dhe ri-shkruar këtu; gabimet janë tona.

Më shumë në Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs