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To save laymen a google: A complete graph is one where all the nodes are connected. A planar graph is one that can be draw in the plane without two lines intersecting.

Notably one you can draw on either a flat piece of paper or a sphere. On a torus, you need up to 7 colors.

Yep, and the proof that any graph that can be drawn without crossings on the torus can be properly colored with no more than 7 colors is vastly easier than the case for graphs drawn on the sphere/plane.

Yeah, any graph of a sphere can be easily deformed to fit on a plane, with something like stereographic projection

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