Polychromatic colorings of n-dimensional guillotine-partitions

Abstract

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. © 2008 Springer-Verlag Berlin Heidelberg.