2022/12/06 by Biswas, Ranita, di Montesano, Sebastiano Cultrera, Draganov, Ondřej +2 · 1 citation
#Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.03121
Given a locally finite set A ⊆ ℝd and a coloring χ\colon A → \0,1,…,s\, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in ℝs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n = #A, and the coloring is random, then the chromatic Delaunay mosaic has O(n^\lceild/2\rceil) cells in expectation. In contrast, for Delone sets and Poisson point processes in ℝd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in ℝ2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.