{"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}
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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 $$\lfloor 3n-\sqrt{12n-3}\rfloor $$ ⌊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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