Colorability in orthogonal graph drawing

This paper studies the question: What is the maximum integer k(b,n) such that every k(b,n)-colorable graph has a b-bend n-dimensional orthogonal box drawing? We give an exact answer for the orthogonal line drawing in all dimensions and for the 3-dimensional rectangle visibility representation. We present an upper and lower bound for the 3-dimensional orthogonal drawing by rectangles and general boxes. Particularly, we improve the best known upper bound for the 3-dimensional orthogonal box drawing from 183 to 42 and the lower bound from 3 to 22.