Graphs obtained from collections of blocks

• Main Authors: Colton Magnant, Pouria Salehi Nowbandegani, Hua Wang
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2015-03-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: chromatic number, rectangular blocks
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/81

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.