On coloring box graphs

Abstract

We consider the chromatic number of a family of graphs we call box graphs, which arise from a box complex in $n$-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 $n$-space has an admissible coloring with $n+1$ colors. We show that for box graphs in $n$-space, if the lengths of the boxes in the corresponding box complex take on no more than two values from the set $\{1,2,3\}$, then the box graph is $3$-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 $3$-colorable.