{"title":"周长至少为8的图是b连续的","authors":"Allen Ibiapina, Ana Silva","doi":"10.1016/j.entcs.2019.08.059","DOIUrl":null,"url":null,"abstract":"<div><p>A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of <em>G</em> is the set <em>S</em><sub><em>b</em></sub>(<em>G</em>) of integers <em>k</em> such that <em>G</em> has a b-coloring with <em>k</em> colors and <em>b</em>(<em>G</em>) = max<em>S</em><sub><em>b</em></sub>(<em>G</em>) is the b-chromatic number of <em>G</em>. A graph is b-continous if <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mo>[</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>]</mo><mo>∩</mo><mi>Z</mi></math></span>. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph <em>G</em> with girth at least 7 contains the integers between 2<em>χ</em>(<em>G</em>) and <em>b</em>(<em>G</em>). This generalizes a previous result by Linhares-Sales and Silva (2017), and tells that graphs with girth at least 7 are, in a way, almost b-continuous.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":"346 ","pages":"Pages 677-684"},"PeriodicalIF":0.0000,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.059","citationCount":"0","resultStr":"{\"title\":\"Graphs with Girth at Least 8 are b-continuous\",\"authors\":\"Allen Ibiapina, Ana Silva\",\"doi\":\"10.1016/j.entcs.2019.08.059\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of <em>G</em> is the set <em>S</em><sub><em>b</em></sub>(<em>G</em>) of integers <em>k</em> such that <em>G</em> has a b-coloring with <em>k</em> colors and <em>b</em>(<em>G</em>) = max<em>S</em><sub><em>b</em></sub>(<em>G</em>) is the b-chromatic number of <em>G</em>. A graph is b-continous if <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mo>[</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>]</mo><mo>∩</mo><mi>Z</mi></math></span>. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph <em>G</em> with girth at least 7 contains the integers between 2<em>χ</em>(<em>G</em>) and <em>b</em>(<em>G</em>). This generalizes a previous result by Linhares-Sales and Silva (2017), and tells that graphs with girth at least 7 are, in a way, almost b-continuous.</p></div>\",\"PeriodicalId\":38770,\"journal\":{\"name\":\"Electronic Notes in Theoretical Computer Science\",\"volume\":\"346 \",\"pages\":\"Pages 677-684\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.059\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Notes in Theoretical Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1571066119301100\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Computer Science\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Notes in Theoretical Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1571066119301100","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Computer Science","Score":null,"Total":0}
A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of G is the set Sb(G) of integers k such that G has a b-coloring with k colors and b(G) = maxSb(G) is the b-chromatic number of G. A graph is b-continous if . An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph G with girth at least 7 contains the integers between 2χ(G) and b(G). This generalizes a previous result by Linhares-Sales and Silva (2017), and tells that graphs with girth at least 7 are, in a way, almost b-continuous.
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