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The four-color theorem of Appel and Haken says that any map in the plane can be colored with at most four distinct colors so that two regions which share a common boundary segment have distinct colors.

Question. Is there a cell decomposition of the torus into 7 polygons such that each two polygons share at least one side in common?

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You might want to have a look at this:

Torus cut into seven regions, with each pair of regions adjacent somewhere

It is known that at most seven colours are required to colour any map on the torus, and this construction proves that it is the best possible bound.

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Yes. On A torus, $7$ is the maximum. See here: [https://en.wikipedia.org/wiki/Four_color_theorem#Generalizations]

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