{"title":"无短环平面图的Δ+2染色","authors":"Ying Chen, Lan Tao, Li Zhang","doi":"10.1007/s10255-023-1098-8","DOIUrl":null,"url":null,"abstract":"<div><p>A coloring of graph <i>G</i> is an <i>injective coloring</i> if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The <i>injective chromatic number</i> χ<sub><i>i</i></sub>(<i>G</i>) of <i>G</i> is the least integer <i>k</i> such that <i>G</i> has an injective <i>k</i>-coloring. In this paper, we prove that (1) if <i>G</i> is a planar graph with girth <i>g</i> ≥ 6 and maximum degree Δ ≥ 7, then <i>χ</i><sub><i>i</i></sub>(<i>G</i>) ≤ Δ + 2; (2) if <i>G</i> is a planar graph with Δ ≥ 24 and without 3,4,7-cycles, then <i>χ</i><sub><i>i</i></sub>(<i>G</i>) ≤ Δ + 2.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Injective Δ+2 Coloring of Planar Graph Without Short Cycles\",\"authors\":\"Ying Chen, Lan Tao, Li Zhang\",\"doi\":\"10.1007/s10255-023-1098-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A coloring of graph <i>G</i> is an <i>injective coloring</i> if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The <i>injective chromatic number</i> χ<sub><i>i</i></sub>(<i>G</i>) of <i>G</i> is the least integer <i>k</i> such that <i>G</i> has an injective <i>k</i>-coloring. In this paper, we prove that (1) if <i>G</i> is a planar graph with girth <i>g</i> ≥ 6 and maximum degree Δ ≥ 7, then <i>χ</i><sub><i>i</i></sub>(<i>G</i>) ≤ Δ + 2; (2) if <i>G</i> is a planar graph with Δ ≥ 24 and without 3,4,7-cycles, then <i>χ</i><sub><i>i</i></sub>(<i>G</i>) ≤ Δ + 2.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-023-1098-8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-023-1098-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Injective Δ+2 Coloring of Planar Graph Without Short Cycles
A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number χi(G) of G is the least integer k such that G has an injective k-coloring. In this paper, we prove that (1) if G is a planar graph with girth g ≥ 6 and maximum degree Δ ≥ 7, then χi(G) ≤ Δ + 2; (2) if G is a planar graph with Δ ≥ 24 and without 3,4,7-cycles, then χi(G) ≤ Δ + 2.
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