{"title":"Coloring lines and Delaunay graphs with respect to boxes","authors":"Tomon, István","doi":"10.1002/rsa.21193","DOIUrl":null,"url":null,"abstract":"The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of $n$ lines in $\\mathbb{R}^3$ whose intersection graph is triangle-free of chromatic number $\\Omega(n^{1/15})$. This improves the previously best known bound $\\Omega(\\log\\log n)$ by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of $n$ points in $\\mathbb{R}^d$, whose Delaunay graph with respect to axis-parallel boxes has independence number at most $n\\cdot (\\log n)^{-(d-1)/2+o(1)}$. This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"21 1","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21193","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 1
Abstract
The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of $n$ lines in $\mathbb{R}^3$ whose intersection graph is triangle-free of chromatic number $\Omega(n^{1/15})$. This improves the previously best known bound $\Omega(\log\log n)$ by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of $n$ points in $\mathbb{R}^d$, whose Delaunay graph with respect to axis-parallel boxes has independence number at most $n\cdot (\log n)^{-(d-1)/2+o(1)}$. This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.
期刊介绍:
It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness.
Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.