Graphs obtained from collections of blocks

Colton Magnant (Georgia Southern University)
Pouria Salehi Nowbandegani
Hua Wang

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 3, No 1 (2015): Electronic Journal of Graph Theory and Applications

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 $d \geq 3$, 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.