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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?

Dabar tu Joks skaičiuotuvas nenusistato, bet jo dalys yra skaičiuojamos. Pabandykite vieną žemiau arba įveskite savo.

Išlaikyti savo darbą

Nemokama paskyra prideda pastabas apie kiekvieną pamoką, įrašą apie tai, ką baigėte, savo išspręstas problemas vienoje vietoje ir mokytoją, kurį galite paklausti apie šį puslapį. Pati matematika yra atvira visiems, kurie yra ar nėra užsiregistravę.

Prisijungti Prisijungti

Čia naudojami simboliai

Bakstelėkite visus simbolius, kad būtų galima apibrėžti visą apibrėžimą, paveikslėlį ir ką reiškia kiekviena raidė.

Klausimai, kuriuos klausia žmonės

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.

Šio puslapio dalys pritaikytos nuo Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Čia yra įtikinamų ir iš naujo paaiškintų klaidų.

Daugiau informacijos 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