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Colton Magnant
Georgia Southern University
Electrical & Electronics Engineering

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Graphs obtained from collections of blocks Colton Magnant; 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
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2015.3.1.6

Abstract

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.
Co-Authors Hua Wang Pouria Salehi Nowbandegani