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Encyclopedia > Three cottage problem

The three cottage problem is a problem in mathematical graph theory:

Suppose there are three cottages that each need to be connected to the gas, water, and electric companies. Is there a way to do so without any of the lines crossing each other?

In more formal terms, this asks whether the complete bipartite graph K3,3 is planar. Kazimierz Kuratowski proved in 1930 that K3,3 is nonplanar, and thus that the three cottage problem has no solution.


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Graph theory (1909 words)
One of the most famous and productive problems of graph theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?".
This problem was first posed by Francis Guthrie in 1852 and the first written record of this problem is a letter of De Morgan addressed to Hamilton the same year.
Covering problems are specific instances of subgraph-finding problems, and they tend to be closely related to the clique problem or the independent set problem.
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