Euler Characteristic

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The Euler Characteristic $\chi$ was classically defined for polyhedra, according to the formula:

$\chi=V-E+F$

and is equal to 2 on the plane (or sphere) and 0 on the torus.

Using the Euler characteristic we can prove that $K_{3,3}$ is nonplanar, and hence the classic three utilities problem (from graph theory) has no solution. (see planar graph)