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Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-convex hexagonal faces meeting 3 at a vertex, and every face meets every other face, so it requires 7 colors.

In 1982 ("Coloring Polyhedral Manifolds"), Barnette proved if you require that every face of the toroidal polyhedron be convex, then the polyhedron can be colored in at most 6 colors, and remarked that all the examples that he knew about could be colored in only 4 colors.

(1) Are there known examples which require 5 colors? 6 colors? Alternately, are there any more recent results on colorability of toroidal polyhedra with convex faces?

(2) Are there any known results about coloring toroidal polyhedra embedded on n-holed tori? Is there a "Szilassi polyhedron analogue" for larger genus? Are there known results if you impose convexity?

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