Graphs obtained from collections of blocks

Given a collection of d-dimensional rectangular solids called blocks, no two of which sharinginterior points, construct a block graph by adding a vertex for each block and an edge if the facesof the two corresponding blocks intersect nontrivially. It is known that if d ≥ 3, such blockgraphs can have arbitrarily large chromatic number. We prove that the chromatic number can bebounded with only a mild restriction on the sizes of the blocks. We also show that block graphs ofblock configurations arising from partitions of d-dimensional hypercubes into sub-hypercubes areat least d-connected. Bounds on the diameter and the hamiltonicity of such block graphs are alsodiscussed.