Isoperimetric Inequalities on Hexagonal Grids
Berit Gru{\ss}ien
TL;DR
This paper investigates isoperimetric inequalities on hexagonal grids, providing tight bounds for edge and vertex boundaries of subsets, with implications for finite and infinite grid structures.
Contribution
It introduces new lower bounds for edge and vertex boundaries on hexagonal grids, including tight bounds for the infinite case.
Findings
Established tight bounds for the infinite hexagonal grid
Provided lower bounds for boundary sizes in finite grids
Extended isoperimetric inequalities to hexagonal grid structures
Abstract
We consider the edge- and vertex-isoperimetric probem on finite and infinite hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we give a lower bound for the number of edges between W and its complement, and lower bounds for the number of vertices in the neighborhood of W and for the number of vertices in the boundary of W. For the infinite hexagonal grid the given bounds are tight.
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Taxonomy
TopicsGraph theory and applications · Point processes and geometric inequalities · Advanced Graph Theory Research