Water, gas, and electricity
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The classical mathematical puzzle known as water, gas, and electricity, the (three) utilities problem, or sometimes the three cottage problem, can be stated as follows:
- Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. Using a third dimension or going through a company or cottage are illegal. Is there a way to do so without any of the lines crossing each other?
[edit] Solution
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Thomsen graph, Utility graph, K3,3 n = 6, m = 9
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The problem is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces. In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 is planar. This graph is often referred to as the utility graph in reference to the problem.[1] The graph is equivalent to the circulant graph Ci6(1,3). Kazimierz Kuratowski proved in 1930 that K3,3 is nonplanar, and thus that the problem has no solution. The solution can be adduced as a consequence of the Jordan curve theorem. The complete statement on the planarity of graphs (the Kuratowski reduction theorem) encompasses this result.
But K3,3 is toroidal, which means it can be embedded on the torus. In terms of the three cottage problem this means the problem can be solved by punching a hole through the plane (or the sphere). This changes the topological properties of the surface and using the hole we can connect the three cottages without crossing lines. An equivalent statement is that the graph genus of the utility graph is one, and therefore it cannot be embedded in a surface of genus less than one. A surface of genus one is equivalent to a torus.
One cannot redraw graphs with non-zero genus in the plane without edge intersections. The three cottage problem is a simplified version of a problem that electronic circuit board designers face. When all of the connections on a circuit board are limited to one side of a board, all nonplanar electronic circuits are impossible. If one is allowed to use a third dimension by using wires or connecting to the second side of the board, all electronic circuit paths are possible.
[edit] References
- ^ Utility Graph from mathworld.wolfram.com
