{"title":"火柴图边数的紧界","authors":"Jérémy Lavollée, Konrad Swanepoel","doi":"10.1007/s00454-023-00530-z","DOIUrl":null,"url":null,"abstract":"Abstract A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices is $$\\lfloor 3n-\\sqrt{12n-3}\\rfloor $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>12</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> . In this paper we prove this conjecture for all $$n\\ge 1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The main geometric ingredient of the proof is an isoperimetric inequality related to L’Huilier’s inequality.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"5 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Tight Bound for the Number of Edges of Matchstick Graphs\",\"authors\":\"Jérémy Lavollée, Konrad Swanepoel\",\"doi\":\"10.1007/s00454-023-00530-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices is $$\\\\lfloor 3n-\\\\sqrt{12n-3}\\\\rfloor $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>12</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> . In this paper we prove this conjecture for all $$n\\\\ge 1$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The main geometric ingredient of the proof is an isoperimetric inequality related to L’Huilier’s inequality.\",\"PeriodicalId\":356162,\"journal\":{\"name\":\"Discrete and Computational Geometry\",\"volume\":\"5 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-08-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete and Computational Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-023-00530-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete and Computational Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00454-023-00530-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
火柴棍图是一种边绘制为单位距离线段的平面图。Harborth在1981年引入了这些图，并推测在n个顶点上的火柴棒图的最大边数是$$\lfloor 3n-\sqrt{12n-3}\rfloor $$⌊3 n - 12 n - 3⌋。本文对所有$$n\ge 1$$ n≥1证明了这个猜想。证明的主要几何成分是一个与L 'Huilier不等式有关的等周不等式。
A Tight Bound for the Number of Edges of Matchstick Graphs
Abstract A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices is $$\lfloor 3n-\sqrt{12n-3}\rfloor $$ ⌊3n-12n-3⌋ . In this paper we prove this conjecture for all $$n\ge 1$$ n≥1 . The main geometric ingredient of the proof is an isoperimetric inequality related to L’Huilier’s inequality.
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