Graphs obtained from collections of blocks - Colton Magnant; Pouria Salehi Nowbandegani; Hua Wang (Electronic Journal of Graph Theory and Applications (EJGTA), 2015)
Given a collection of $d$-dimensional rectangular solids called blocks, no two of which sharing interior points, construct a block graph by adding a vertex for each block and an edge if the faces of the two corresponding blocks intersect nontrivially. It is known that if $d ≥ 3$, such block graphs can have arbitrarily large chromatic number. We prove that the chromatic number can be bounded with only a mild restriction on the sizes of the blocks. We also show that block graphs of block configurations arising from partitions of $d$-dimensional hypercubes into sub-hypercubes are at least $d$-connected. Bounds on the diameter and the hamiltonicity of such block graphs are also discussed.