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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 \(G\) shown in .

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?

Symbols used here

n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.

Questions people ask

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.

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Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

بیشتر در 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