{"title":"圆弧色有向图中的彩虹传递三角形","authors":"Mengyu Duan , Zhiwei Guo , Binlong Li , Shenggui Zhang","doi":"10.1016/j.dam.2025.04.011","DOIUrl":null,"url":null,"abstract":"<div><div>A subdigraph of an arc-colored digraph is <em>rainbow</em> if its all arcs have distinct colors. For two digraphs <span><math><mi>D</mi></math></span> and <span><math><mi>H</mi></math></span>, let <span><math><mrow><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> be the minimum integer such that every arc-colored digraph <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> with <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>≥</mo><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> contains a rainbow copy of <span><math><mi>H</mi></math></span>, where <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> is the number of colors of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span>. Let <span><math><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover></math></span> be the digraph obtained from the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by replacing each edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> with a pair of symmetric arcs <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></math></span>, and let <span><math><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover></math></span> be the transitive triangle. In this paper we determine <span><math><mrow><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> and characterize the corresponding extremal arc-colorings of <span><math><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover></math></span>. Further, we prove that an arc-colored digraph <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> on <span><math><mi>n</mi></math></span> vertices contains a rainbow <span><math><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover></math></span> if <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>≥</mo><mi>a</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>)</mo></mrow><mo>+</mo><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span>. Moreover, if <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>)</mo></mrow><mo>+</mo><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> contains no rainbow <span><math><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover></math></span>’s, then <span><math><mrow><mi>D</mi><mo>≅</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 175-184"},"PeriodicalIF":1.0000,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Rainbow transitive triangles in arc-colored digraphs\",\"authors\":\"Mengyu Duan , Zhiwei Guo , Binlong Li , Shenggui Zhang\",\"doi\":\"10.1016/j.dam.2025.04.011\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>A subdigraph of an arc-colored digraph is <em>rainbow</em> if its all arcs have distinct colors. For two digraphs <span><math><mi>D</mi></math></span> and <span><math><mi>H</mi></math></span>, let <span><math><mrow><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> be the minimum integer such that every arc-colored digraph <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> with <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>≥</mo><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> contains a rainbow copy of <span><math><mi>H</mi></math></span>, where <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> is the number of colors of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span>. Let <span><math><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover></math></span> be the digraph obtained from the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by replacing each edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> with a pair of symmetric arcs <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></math></span>, and let <span><math><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover></math></span> be the transitive triangle. In this paper we determine <span><math><mrow><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> and characterize the corresponding extremal arc-colorings of <span><math><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover></math></span>. Further, we prove that an arc-colored digraph <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> on <span><math><mi>n</mi></math></span> vertices contains a rainbow <span><math><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover></math></span> if <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>≥</mo><mi>a</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>)</mo></mrow><mo>+</mo><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span>. Moreover, if <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>)</mo></mrow><mo>+</mo><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></msup></math></span> contains no rainbow <span><math><mover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>⃗</mo></mover></math></span>’s, then <span><math><mrow><mi>D</mi><mo>≅</mo><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>↔</mo></mrow></mover></mrow></math></span>.</div></div>\",\"PeriodicalId\":50573,\"journal\":{\"name\":\"Discrete Applied Mathematics\",\"volume\":\"370 \",\"pages\":\"Pages 175-184\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-04-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0166218X25001805\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X25001805","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Rainbow transitive triangles in arc-colored digraphs
A subdigraph of an arc-colored digraph is rainbow if its all arcs have distinct colors. For two digraphs and , let be the minimum integer such that every arc-colored digraph with contains a rainbow copy of , where is the number of colors of . Let be the digraph obtained from the complete graph by replacing each edge with a pair of symmetric arcs and , and let be the transitive triangle. In this paper we determine and characterize the corresponding extremal arc-colorings of . Further, we prove that an arc-colored digraph on vertices contains a rainbow if . Moreover, if and contains no rainbow ’s, then .
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The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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