A couple consequences of Euler's formula $V-E+F=2$ for graphs that can make for nice series of homework exercises:

• Predict the structure of Buckminsterfullerenes. Carbon tends to form 5 and 6 member rings, so you can model Buckyballs as convex polyhedra whose faces are all pentagons and hexagons. Using Euler's formula, you can show that all such polyhedra must have exactly $12$ pentagons.
  If you assume the additional fact from chemistry that structures with adjacent pentagonal rings are unstable (due to bond strain along the common edge), you can say that any polyhedron not violating this "isolated pentagon rule" must involve at least $60$ carbon atoms, and describe the structure of the only one with $60$ atoms. This turns out to be exactly the structure of the first buckyball discovered.

• Various results involving the sums of angles. For example, you can use Euler's formula to show that a triangulation of an $n$ sided polygon has $n-2$ triangles, and get the familiar formula for the sum of the angles. From this, a double counting argument, and another application of Euler's formula you can show Descartes rule of angular defect (if the discrepancy at a vertex is $2 \pi$ minus the sum of the angles of each face at that vertex, the sum of the discrepancies of any convex polyhedron is $4 \pi$)