A strong hyperbox-respecting coloring of an n-dimensional hyperbox partition is a coloring of the corners of its hyperboxes with 2 n colors such that any hyperbox has all the colors appearing on its corners. A guillotine-partition is obtained by starting with a single axis-parallel hyperbox and recursively cutting a hyperbox of the partition into two hyperboxes by a hyperplane orthogonal to one of the n axes. We prove that there is a strong hyperbox-respecting coloring of any n-dimensional guillotine-partition. This theorem generalizes the result of Horev et al. [8] who proved the 2-dimensional case. This problem is a special case of the n-dimensional variant of polychromatic colorings. The proof gives an efficient coloring algorithm as well.

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It is shown that every polyhedron in R3 with m edges can be guarded with at most 27 32m edge guards. The bound improves to 5 1