一些完整二方图和三方图的厚度

Pub Date : 2024-11-06 DOI:10.1007/s10255-024-1128-1

Si-wei Hu, Yi-chao Chen

{"title":"一些完整二方图和三方图的厚度","authors":"Si-wei Hu, Yi-chao Chen","doi":"10.1007/s10255-024-1128-1","DOIUrl":null,"url":null,"abstract":"<div>In this paper, we obtain the thickness for some complete k–partite graphs for k = 2, 3. We first compute the thickness of Kn,n+8 by giving a planar decomposition of K4k−1,4k+7 for k ≥ 3. Then, two planar decompositions for K1,g,g(g−1) when g is even and for \\(K_{1,g,{1\\over{2}}(g-1)^{2}}\\) when g is odd are obtained. Using a recursive construction, we also obtain the thickness for some complete tripartite graphs. The results here support the long-standing conjecture that the thickness of Km,n is \\(\\lceil {mn\\over{2(m+n-2)}}\\rceil\\) for any positive integers m, n.</div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Thickness of Some Complete Bipartite and Tripartite Graphs\",\"authors\":\"Si-wei Hu, Yi-chao Chen\",\"doi\":\"10.1007/s10255-024-1128-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>In this paper, we obtain the thickness for some complete k–partite graphs for k = 2, 3. We first compute the thickness of Kn,n+8 by giving a planar decomposition of K4k−1,4k+7 for k ≥ 3. Then, two planar decompositions for K1,g,g(g−1) when g is even and for \\\\(K_{1,g,{1\\\\over{2}}(g-1)^{2}}\\\\) when g is odd are obtained. Using a recursive construction, we also obtain the thickness for some complete tripartite graphs. The results here support the long-standing conjecture that the thickness of Km,n is \\\\(\\\\lceil {mn\\\\over{2(m+n-2)}}\\\\rceil\\\\) for any positive integers m, n.</div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-11-06\",\"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/s10255-024-1128-1\",\"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/s10255-024-1128-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

在本文中，我们得到了 k = 2, 3 时一些完整 k 部分图的厚度。我们首先通过给出 k≥3 时 K4k-1,4k+7 的平面分解来计算 Kn,n+8 的厚度。然后，当 g 为偶数时，得到 K1,g,g(g-1)的两个平面分解；当 g 为奇数时，得到 \(K_{1,g,{1/over{2}}(g-1)^{2}}\) 的两个平面分解。通过递归构造，我们还得到了一些完整三方图的厚度。这里的结果支持了一个存在已久的猜想，即对于任意正整数 m、n，Km,n 的厚度都是\(\lceil {mn\over{2(m+n-2)}}\rceil\) 。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

The Thickness of Some Complete Bipartite and Tripartite Graphs

In this paper, we obtain the thickness for some complete k–partite graphs for k = 2, 3. We first compute the thickness of K_n,n+8 by giving a planar decomposition of K_4k−1,4k+7 for k ≥ 3. Then, two planar decompositions for K_1,g,g(g−1) when g is even and for \(K_{1,g,{1\over{2}}(g-1)^{2}}\) when g is odd are obtained. Using a recursive construction, we also obtain the thickness for some complete tripartite graphs. The results here support the long-standing conjecture that the thickness of K_m,n is \(\lceil {mn\over{2(m+n-2)}}\rceil\) for any positive integers m, n.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助