Grid intersection graphs and boxicity

A graph has hyuiciry k if k is the smallest integer such that G is an intersection graph of k-dimensional boxes in a &-dimensional space (where the sides of the boxes are parallel to the coordinate axis). A graph has grid dimension k if k is the smallest integer such that G is an intersection graph of k-dimensional boxes (parallel to the coordinate axis) in a (k+ I)-dimensional space. We prove that all bipartite graphs with boxicity two. have grid dimensions one, that is, they can be represented as intersection graphs of horizontal and vertical intervals in the plane. We also introduce some inequalities for the grid dimension of a graph, and discuss extremal graphs with large grid dimensions.