{"title":"无 (P5, HVN) 图形的色度数","authors":"Yian Xu","doi":"10.1007/s10255-024-1029-3","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a graph. We use <i>χ</i>(<i>G</i>) and <i>ω</i>(<i>G</i>) to denote the chromatic number and clique number of <i>G</i> respectively. A <i>P</i><sub>5</sub> is a path on 5 vertices, and an HVN is a <i>K</i><sub>4</sub> together with one more vertex which is adjacent to exactly two vertices of <i>K</i><sub>4</sub>. Combining with some known result, in this paper we show that if <i>G</i> is (<i>P</i><sub>5</sub>, <i>HVN</i>)-free, then <i>χ</i>(<i>G</i>) ≤ max{min{16, <i>ω</i>(<i>G</i>) + 3}, <i>ω</i>(<i>G</i>) + 1}. This upper bound is almost sharp.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 4","pages":"1098 - 1110"},"PeriodicalIF":0.9000,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Chromatic Number of (P5, HVN)-free Graphs\",\"authors\":\"Yian Xu\",\"doi\":\"10.1007/s10255-024-1029-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>G</i> be a graph. We use <i>χ</i>(<i>G</i>) and <i>ω</i>(<i>G</i>) to denote the chromatic number and clique number of <i>G</i> respectively. A <i>P</i><sub>5</sub> is a path on 5 vertices, and an HVN is a <i>K</i><sub>4</sub> together with one more vertex which is adjacent to exactly two vertices of <i>K</i><sub>4</sub>. Combining with some known result, in this paper we show that if <i>G</i> is (<i>P</i><sub>5</sub>, <i>HVN</i>)-free, then <i>χ</i>(<i>G</i>) ≤ max{min{16, <i>ω</i>(<i>G</i>) + 3}, <i>ω</i>(<i>G</i>) + 1}. This upper bound is almost sharp.</p></div>\",\"PeriodicalId\":6951,\"journal\":{\"name\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"volume\":\"40 4\",\"pages\":\"1098 - 1110\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-11-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-024-1029-3\",\"RegionNum\":4,\"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":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1029-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Let G be a graph. We use χ(G) and ω(G) to denote the chromatic number and clique number of G respectively. A P5 is a path on 5 vertices, and an HVN is a K4 together with one more vertex which is adjacent to exactly two vertices of K4. Combining with some known result, in this paper we show that if G is (P5, HVN)-free, then χ(G) ≤ max{min{16, ω(G) + 3}, ω(G) + 1}. This upper bound is almost sharp.
期刊介绍：
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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