{"title":"偶数情况下的座位偶数问题","authors":"","doi":"10.1016/j.disc.2024.114182","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we consider the seating couples problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer <span><math><mi>v</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span> and a list <em>L</em> containing <em>n</em> positive integers not exceeding <em>n</em>, is it always possible to find a perfect matching of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> whose list of edge-lengths is <em>L</em>? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with <em>v</em>. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers <span><math><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>x</mi></math></span>, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The seating couples problem in the even case\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114182\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we consider the seating couples problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer <span><math><mi>v</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span> and a list <em>L</em> containing <em>n</em> positive integers not exceeding <em>n</em>, is it always possible to find a perfect matching of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> whose list of edge-lengths is <em>L</em>? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with <em>v</em>. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers <span><math><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>x</mi></math></span>, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24003133\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003133","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this paper we consider the seating couples problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer and a list L containing n positive integers not exceeding n, is it always possible to find a perfect matching of whose list of edge-lengths is L? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with v. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers , each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.