2013/09/12 by Ellen Veomett, Veomett, Ellen
Computer Science · Engineering · #05C12 #05C40 #05C62 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mechanical Behavior of Composites #VLSI and FPGA Design Techniques
paper · pdf · doi:10.48550/arxiv.1309.3251
openalex publication_date 2013/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the family of graphs whose vertex set is ℤn where two vertices are connected by an edge when their ℓ_∞-distance is 1. Towards an edge isoperimetric inequality for this graph, we calculate the edge boundary of any finite set S ⊂ ℤn. This boundary calculation leads to a desire to show that a set with optimal edge boundary has no ``gaps'' in any direction ε∈ \-1,0,1\n, ε\not=0. We show that one can find a set with optimal edge boundary that does not have gaps in any direction ei (or -ei) where ei is the standard basis vector.