On coloring box graphs

We consider the chromatic number of a family of graphs we call box graphs, which arise from a box complex in nn-space. It is straightforward to show that any box graph in the plane has an admissible coloring with three colors, and that any box graph in nn-space has an admissible coloring with n+1n+1 colors. We show that for box graphs in nn-space, if the lengths of the boxes in the corresponding box complex take on no more than two values from the set {1,2,3}{1,2,3}, then the box graph is 33-colorable, and for some graphs three colors are required. We also show that box graphs in 3-space which do not have cycles of length four (which we call “string complexes”) are 33-colorable.