{"title":"多边形短链的不连续图","authors":"János Pach , Gábor Tardos , Géza Tóth","doi":"10.1016/j.jctb.2023.08.008","DOIUrl":null,"url":null,"abstract":"<div><p>The <em>disjointness graph</em> of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph <em>G</em> of any system of segments in the plane is <em>χ-bounded</em>, that is, its chromatic number <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is upper bounded by a function of its clique number <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2<em>-way infinite</em> polygonal chains of length 3 is <em>χ</em>-bounded: for every such graph <em>G</em>, we have <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msup><mrow><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 29-43"},"PeriodicalIF":1.2000,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Disjointness graphs of short polygonal chains\",\"authors\":\"János Pach , Gábor Tardos , Géza Tóth\",\"doi\":\"10.1016/j.jctb.2023.08.008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The <em>disjointness graph</em> of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph <em>G</em> of any system of segments in the plane is <em>χ-bounded</em>, that is, its chromatic number <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is upper bounded by a function of its clique number <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2<em>-way infinite</em> polygonal chains of length 3 is <em>χ</em>-bounded: for every such graph <em>G</em>, we have <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msup><mrow><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p></div>\",\"PeriodicalId\":54865,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series B\",\"volume\":\"164 \",\"pages\":\"Pages 29-43\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series B\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895623000679\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895623000679","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number is upper bounded by a function of its clique number .
Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have .
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.