Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.