On the Size of Chromatic Delaunay Mosaics

[2212.03121] On the Size of Chromatic Delaunay Mosaics Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2212.03121 (math) [Submitted on 6 Dec 2022] Title: On the Size of Chromatic Delaunay Mosaics Authors: Ranita Biswas , Sebastiano Cultrera di Montesano , Ondřej Draganov , Herbert Edelsbrunner , Morteza Saghafian View PDF HTML (experimental) Abstract: Given a locally finite set $A \subseteq \mathbb{R}^d$ and a coloring $\chi \colon A \to \{0,1,\ldots,s\}$, we introduce the chromatic Delaunay mosaic of $\chi$, which is a Delaunay mosaic in $\mathbb{R}^{s+d}$ 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 $O(n^{\lceil{d/2}\rceil})$ cells in expectation. In contrast, for Delone sets and Poisson point processes in $\mathbb{R}^d$, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in $\mathbb{R}^2$ 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. Comments: 15 pages, 4 figures Subjects: Combinatorics (math.CO) ; Computational Geometry (cs.CG) Cite as: arXiv:2212.03121 [math.CO] (or arXiv:2212.03121v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2212.03121 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s00454-025-00778-7 Focus to learn more DOI(s) linking to related resources Submission history From: Ondřej Draganov [ view email ] [v1] Tue, 6 Dec 2022 16:37:01 UTC (563 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2022-12 Change to browse by: cs cs.CG math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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