{"title":"Dimension is polynomial in height for posets with planar cover graphs","authors":"Jakub Kozik , Piotr Micek , William T. Trotter","doi":"10.1016/j.jctb.2023.10.009","DOIUrl":null,"url":null,"abstract":"<div><p>We show that height <em>h</em><span> posets that have planar cover graphs have dimension </span><span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span>. Previously, the best upper bound was <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></msup></math></span><span>. Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes </span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> as a minor.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"165 ","pages":"Pages 164-196"},"PeriodicalIF":1.2000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895623000916","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 8
Abstract
We show that height h posets that have planar cover graphs have dimension . Previously, the best upper bound was . Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes as a minor.
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
