{"title":"单平面图形的顶点有性","authors":"Dongdong Zhang, Juan Liu, Yongjie Li, Hehua Yang","doi":"10.1007/s00373-024-02820-6","DOIUrl":null,"url":null,"abstract":"<p>The vertex arboricity <i>a</i>(<i>G</i>) of a graph <i>G</i> is the minimum number of colors required to color the vertices of <i>G</i> such that no cycle is monochromatic. A graph <i>G</i> is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we proved that every 1-planar graph without 5-cycles has minimum degree at most 5; Every 1-planar graph of girth at least 7 has minimum degree at most 3. The following conclusions can be obtained by combining the existing conclusions and our proofs: if <i>G</i> is a 1-planar graph without 5-cycles, then <span>\\(a(G)\\le 3\\)</span>; if <i>G</i> is a 1-planar graph with <span>\\(g(G)\\ge 7\\)</span>, then <span>\\(a(G)\\le 2\\)</span>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Vertex Arboricity of 1-Planar Graphs\",\"authors\":\"Dongdong Zhang, Juan Liu, Yongjie Li, Hehua Yang\",\"doi\":\"10.1007/s00373-024-02820-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The vertex arboricity <i>a</i>(<i>G</i>) of a graph <i>G</i> is the minimum number of colors required to color the vertices of <i>G</i> such that no cycle is monochromatic. A graph <i>G</i> is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we proved that every 1-planar graph without 5-cycles has minimum degree at most 5; Every 1-planar graph of girth at least 7 has minimum degree at most 3. The following conclusions can be obtained by combining the existing conclusions and our proofs: if <i>G</i> is a 1-planar graph without 5-cycles, then <span>\\\\(a(G)\\\\le 3\\\\)</span>; if <i>G</i> is a 1-planar graph with <span>\\\\(g(G)\\\\ge 7\\\\)</span>, then <span>\\\\(a(G)\\\\le 2\\\\)</span>.</p>\",\"PeriodicalId\":12811,\"journal\":{\"name\":\"Graphs and Combinatorics\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Graphs and Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00373-024-02820-6\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Graphs and Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02820-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
图 G 的顶点可着色性 a(G) 是指给图 G 的顶点着色时所需的最少颜色数,这样就不会出现单色循环。如果一个图 G 可以在平面上绘制,且每条边最多有一个交叉点,那么它就是 1-平面图。结合已有的结论和我们的证明,可以得到以下结论:如果 G 是一个没有 5 个循环的 1-planar 图,则 \(a(G)\le 3\); 如果 G 是一个有 \(g(G)\ge 7\) 的 1-planar 图,则 \(a(G)\le 2\).
The vertex arboricity a(G) of a graph G is the minimum number of colors required to color the vertices of G such that no cycle is monochromatic. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we proved that every 1-planar graph without 5-cycles has minimum degree at most 5; Every 1-planar graph of girth at least 7 has minimum degree at most 3. The following conclusions can be obtained by combining the existing conclusions and our proofs: if G is a 1-planar graph without 5-cycles, then \(a(G)\le 3\); if G is a 1-planar graph with \(g(G)\ge 7\), then \(a(G)\le 2\).
期刊介绍:
Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.