2010/10/21 by Tom Eccles, Eccles, Tom
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems #math.CO
paper · pdf · doi:10.48550/arxiv.1010.4578
12 pages
arxiv created 2010/10/21 · openalex publication_date 2010/10/21 · arxiv updated 2010/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For integers m1,...,md>0 and a cuboid M=[0,m1]× ... × [0,md]⊂ ℝd, a brick of M is a closed cuboid whose vertices have integer coordinates. A set H of bricks in M is a system of brick islands if for each pair of bricks in H one contains the other or they are disjoint. Such a system is maximal if it cannot be extended to a larger system of brick islands. Extending the work of Lengvárszky, we show that the minimum size of a maximal system of brick islands in M is ∑i=1d mi - (d-1). Also, in a cube C=[0,m]d we define the corresponding notion of a system of cubic islands, and prove bounds on the sizes of maximal systems of cubic islands.