James Davies, Tomasz Krawczyk, Rose McCarty, Bartosz Walczak
{"title":"Grounded L-Graphs Are Polynomially $$chi $$ -Bounded","authors":"James Davies, Tomasz Krawczyk, Rose McCarty, Bartosz Walczak","doi":"10.1007/s00454-023-00592-z","DOIUrl":"https://doi.org/10.1007/s00454-023-00592-z","url":null,"abstract":"<p>A <i>grounded L-graph</i> is the intersection graph of a collection of “L” shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number <span>(omega )</span> has chromatic number at most <span>(17omega ^4)</span>. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially <span>(chi )</span>-bounded. We also survey <span>(chi )</span>-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some New Results on Geometric Transversals","authors":"Otfried Cheong, Xavier Goaoc, Andreas F. Holmsen","doi":"10.1007/s00454-023-00573-2","DOIUrl":"https://doi.org/10.1007/s00454-023-00573-2","url":null,"abstract":"<p>We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Bárány and Kalai regarding weak <span>(varepsilon )</span>-nets for <i>k</i>-flats and convex sets in <span>(mathbb {R}^d)</span>, and we prove a conjecture of Arocha, Bracho, and Montejano regarding a colorful version of the Goodman–Pollack–Wenger transversal theorem. We also investigate the connected components of the space of line transversals to pairwise disjoint convex sets in <span>(mathbb {R}^3)</span>, and we extend a theorem of Karasev and Montejano regarding colorful intersections and <i>k</i>-transversals.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"130 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Visible-Volume Function of a Set of Cameras is Continuous, Piecewise Rational, Locally Lipschitz, and Semi-Algebraic in All Dimensions","authors":"Jörg Rambau","doi":"10.1007/s00454-023-00541-w","DOIUrl":"https://doi.org/10.1007/s00454-023-00541-w","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":"1038 - 1058"},"PeriodicalIF":0.8,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43661006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Delaunay and Regular Triangulations as Lexicographic Optimal Chains","authors":"D. Cohen-Steiner, A. Lieutier, J. Vuillamy","doi":"10.1007/s00454-023-00485-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00485-1","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":"1 - 50"},"PeriodicalIF":0.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45486294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tropical Carathéodory with Matroids.","authors":"Georg Loho, Raman Sanyal","doi":"10.1007/s00454-022-00446-0","DOIUrl":"https://doi.org/10.1007/s00454-022-00446-0","url":null,"abstract":"<p><p>Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 1","pages":"139-155"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9805987/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10494381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}