{"title":"关于格着色的最优性","authors":"Y. Ben-Haim, T. Etzion","doi":"10.1137/S0895480104439589","DOIUrl":null,"url":null,"abstract":"For z/sub 1/, z/sub 2/, z/sub 3/ /spl isin/ Z/sup 2/, the tristance d/sub 3/(z/sub 1/, z/sub 2/, z/sub 3/) is a generalization of the L/sub 1/-distance on Z/sup 2/ to a quality that reflects the relative dispersion of three points rather than two. We prove that at least 3k/sup 2/ colors are required to color the points of Z/sup 2/, such that the tristance between any three distinct points, colored with the same color, is at least 4k. We also prove that 3k/sup 2/+3k+1 colors are required if the tristance is at least 4k+2. For the first case we show an infinite family of colorings with 3k/sup 2/ colors and conjecture that these are the only colorings with 3k/sup 2/ colors.","PeriodicalId":269907,"journal":{"name":"International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2004-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"On the optimality of coloring with a lattice\",\"authors\":\"Y. Ben-Haim, T. Etzion\",\"doi\":\"10.1137/S0895480104439589\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For z/sub 1/, z/sub 2/, z/sub 3/ /spl isin/ Z/sup 2/, the tristance d/sub 3/(z/sub 1/, z/sub 2/, z/sub 3/) is a generalization of the L/sub 1/-distance on Z/sup 2/ to a quality that reflects the relative dispersion of three points rather than two. We prove that at least 3k/sup 2/ colors are required to color the points of Z/sup 2/, such that the tristance between any three distinct points, colored with the same color, is at least 4k. We also prove that 3k/sup 2/+3k+1 colors are required if the tristance is at least 4k+2. For the first case we show an infinite family of colorings with 3k/sup 2/ colors and conjecture that these are the only colorings with 3k/sup 2/ colors.\",\"PeriodicalId\":269907,\"journal\":{\"name\":\"International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2004-06-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/S0895480104439589\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/S0895480104439589","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
For z/sub 1/, z/sub 2/, z/sub 3/ /spl isin/ Z/sup 2/, the tristance d/sub 3/(z/sub 1/, z/sub 2/, z/sub 3/) is a generalization of the L/sub 1/-distance on Z/sup 2/ to a quality that reflects the relative dispersion of three points rather than two. We prove that at least 3k/sup 2/ colors are required to color the points of Z/sup 2/, such that the tristance between any three distinct points, colored with the same color, is at least 4k. We also prove that 3k/sup 2/+3k+1 colors are required if the tristance is at least 4k+2. For the first case we show an infinite family of colorings with 3k/sup 2/ colors and conjecture that these are the only colorings with 3k/sup 2/ colors.
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