--- _id: '9468' abstract: - lang: eng text: "Motivated by the successful application of geometry to proving the Harary--Hill conjecture for “pseudolinear” drawings of $K_n$, we introduce “pseudospherical” drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\\mathbb{S}^2$ in which the vertices of $G$ are represented as points---no three on a great circle---and the edges of $G$ are shortest-arcs in $\\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $\\gamma_e$ such that the only vertices in $\\gamma_e$ are the ends of $e$; (2) if $e\\ne f$, then $\\gamma_e\\cap\\gamma_f$ has precisely two crossings; and (3) if $e\\ne f$, then $e$ intersects $\\gamma_f$ at most once, in either a crossing or an end of $e$. We use properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that for the complete graph, properties (1)--(3) are equivalent to the same three properties but with “precisely two crossings” in (2) replaced by “at most two crossings.” The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs (coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.\r\n" article_processing_charge: No article_type: original author: - first_name: Alan M full_name: Arroyo Guevara, Alan M id: 3207FDC6-F248-11E8-B48F-1D18A9856A87 last_name: Arroyo Guevara orcid: 0000-0003-2401-8670 - first_name: R. Bruce full_name: Richter, R. Bruce last_name: Richter - first_name: Matthew full_name: Sunohara, Matthew last_name: Sunohara citation: ama: Arroyo Guevara AM, Richter RB, Sunohara M. Extending drawings of complete graphs into arrangements of pseudocircles. SIAM Journal on Discrete Mathematics. 2021;35(2):1050-1076. doi:10.1137/20M1313234 apa: Arroyo Guevara, A. M., Richter, R. B., & Sunohara, M. (2021). Extending drawings of complete graphs into arrangements of pseudocircles. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/20M1313234 chicago: Arroyo Guevara, Alan M, R. Bruce Richter, and Matthew Sunohara. “Extending Drawings of Complete Graphs into Arrangements of Pseudocircles.” SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics, 2021. https://doi.org/10.1137/20M1313234. ieee: A. M. Arroyo Guevara, R. B. Richter, and M. Sunohara, “Extending drawings of complete graphs into arrangements of pseudocircles,” SIAM Journal on Discrete Mathematics, vol. 35, no. 2. Society for Industrial and Applied Mathematics, pp. 1050–1076, 2021. ista: Arroyo Guevara AM, Richter RB, Sunohara M. 2021. Extending drawings of complete graphs into arrangements of pseudocircles. SIAM Journal on Discrete Mathematics. 35(2), 1050–1076. mla: Arroyo Guevara, Alan M., et al. “Extending Drawings of Complete Graphs into Arrangements of Pseudocircles.” SIAM Journal on Discrete Mathematics, vol. 35, no. 2, Society for Industrial and Applied Mathematics, 2021, pp. 1050–76, doi:10.1137/20M1313234. short: A.M. Arroyo Guevara, R.B. Richter, M. Sunohara, SIAM Journal on Discrete Mathematics 35 (2021) 1050–1076. date_created: 2021-06-06T22:01:30Z date_published: 2021-05-20T00:00:00Z date_updated: 2023-08-08T13:58:12Z day: '20' department: - _id: UlWa doi: 10.1137/20M1313234 ec_funded: 1 external_id: arxiv: - '2001.06053' isi: - '000674142200022' intvolume: ' 35' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2001.06053 month: '05' oa: 1 oa_version: Preprint page: 1050-1076 project: - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships publication: SIAM Journal on Discrete Mathematics publication_identifier: issn: - '08954801' publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: Extending drawings of complete graphs into arrangements of pseudocircles type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 35 year: '2021' ... --- _id: '312' abstract: - lang: eng text: Motivated by biological questions, we study configurations of equal spheres that neither pack nor cover. Placing their centers on a lattice, we define the soft density of the configuration by penalizing multiple overlaps. Considering the 1-parameter family of diagonally distorted 3-dimensional integer lattices, we show that the soft density is maximized at the FCC lattice. acknowledgement: This work was partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics,” through grant I02979-N35 of the Austrian Science Fund (FWF). article_processing_charge: No article_type: original author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham citation: ama: Edelsbrunner H, Iglesias Ham M. On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. 2018;32(1):750-782. doi:10.1137/16M1097201 apa: Edelsbrunner, H., & Iglesias Ham, M. (2018). On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. Society for Industrial and Applied Mathematics . https://doi.org/10.1137/16M1097201 chicago: Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” SIAM J Discrete Math. Society for Industrial and Applied Mathematics , 2018. https://doi.org/10.1137/16M1097201. ieee: H. Edelsbrunner and M. Iglesias Ham, “On the optimality of the FCC lattice for soft sphere packing,” SIAM J Discrete Math, vol. 32, no. 1. Society for Industrial and Applied Mathematics , pp. 750–782, 2018. ista: Edelsbrunner H, Iglesias Ham M. 2018. On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. 32(1), 750–782. mla: Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” SIAM J Discrete Math, vol. 32, no. 1, Society for Industrial and Applied Mathematics , 2018, pp. 750–82, doi:10.1137/16M1097201. short: H. Edelsbrunner, M. Iglesias Ham, SIAM J Discrete Math 32 (2018) 750–782. date_created: 2018-12-11T11:45:46Z date_published: 2018-03-29T00:00:00Z date_updated: 2023-09-13T09:34:38Z day: '29' department: - _id: HeEd doi: 10.1137/16M1097201 external_id: isi: - '000428958900038' intvolume: ' 32' isi: 1 issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: http://pdfs.semanticscholar.org/d2d5/6da00fbc674e6a8b1bb9d857167e54200dc6.pdf month: '03' oa: 1 oa_version: Submitted Version page: 750 - 782 project: - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: SIAM J Discrete Math publication_identifier: issn: - '08954801' publication_status: published publisher: 'Society for Industrial and Applied Mathematics ' publist_id: '7553' quality_controlled: '1' scopus_import: '1' status: public title: On the optimality of the FCC lattice for soft sphere packing type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 32 year: '2018' ...