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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?
ngoku Akukho calculator icwangcisa le nto, kodwa iinxalenye zayo zibalaseleyo. Zama enye ezantsi, okanye ubhale yakho.
I akhawunti ekhululekileyo idibanisa amaphetshana kwi ncwadi nganye, irekhodi lento ogqibe ngayo, iingxaki zakho ezisoliweyo kwindawo enye, nomfundi onokuthi ubuze malunga nale phepha. IiMathematiki ngokwazo zivuliwe kubo bonke, bangeniswe okanye hayi.
Bhalisa Igama elithileIimpawu ezisetyenziswa apha
Nqakraza nasiphi na isibonakaliso sokuqonda okupheleleyo, umfanekiso, nokuba iileta zonke zithetha ntoni.
Imibuzo abantu bebuza
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.
Iindawo zephepha zitshintshiwe Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Igqityiwe kwaye iphinde yacaciswa apha; iimpazamo zethu.
IiNkqubo 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