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I'm looking for some nice theories or just exercises, with both geometrical aspects and graph theoretics aspects.

Example may include for instance the 4-color theorem or Euler characteristics, maybe Königsberg's bridges (although this last one is almost purely graph theoretic). Ideally I'm looking for material that is not too advanced, like first year of university.

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The five platonic solids and the dual of each one (another platonic solid). – coffeebelly Jul 13 '14 at 18:50
yes, the topological proof using Euler characteristic is perfectly in the theme. – Denis Jul 13 '14 at 19:01
Most of the things I keep thinking of are topological in nature. Like Poincare cubes or gluing flat sheets to get compact surfaces. – coffeebelly Jul 13 '14 at 19:07
Anything relating Euler characteristic and triangulation. For instance, the classification of compact surfaces. – user40276 Jul 13 '14 at 19:13
Maybe proofs showing $k_{3,3}$ and $k_5$ can not be imbedded in the plane. – coffeebelly Jul 13 '14 at 19:17

Oriented matroids may be a link between geometry and graph theory.

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