Fei-Huang Chang, Ma-Lian Chia, Shih-Ang Jiang, David Kuo, Jing-Ho Yan
{"title":"路径和循环的笛卡尔乘积的 n 折 L(2, 1)标记","authors":"Fei-Huang Chang, Ma-Lian Chia, Shih-Ang Jiang, David Kuo, Jing-Ho Yan","doi":"10.1007/s10878-024-01119-9","DOIUrl":null,"url":null,"abstract":"<p>For two sets of nonnegative integers <i>A</i> and <i>B</i>, the distance between these two sets, denoted by <i>d</i>(<i>A</i>, <i>B</i>), is defined by <span>\\(d(A,B)=\\min \\{|a-b|:a\\in A,b\\in B\\}\\)</span>. For a positive integer <i>n</i>, let <span>\\(S_{n}\\)</span> denote the family <span>\\( \\{X:X\\subseteq {\\mathbb {N}} \\cup \\{0\\},|X|=n\\}\\)</span>. Given a graph <i>G</i> and positive integers <i>n</i>, <i>p</i> and <i>q</i>, an <i>n</i>-fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling of <i>G</i> is a function <span>\\(f:V(G)\\rightarrow S_{n} \\)</span> satisfies <span>\\(d(f(u),f(v))\\ge p\\)</span> if <span>\\(d_{G}(u,v)=1\\)</span>, and <span>\\( d(f(u),f(v))\\ge q\\)</span> if <span>\\(d_{G}(u,v)=2\\)</span>. An <i>n</i>-fold <i>k</i>-<i>L</i>(<i>p</i>, <i>q</i>)-labeling <i>f</i> of <i>G</i> is an <i>n</i>-fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling of <i>G</i> with the property that <span>\\(\\max \\{a:a\\in \\bigcup _{u\\in V(G)}f(u)\\}\\le k\\)</span>. The smallest number <i>k</i> to guarantee that <i>G</i> has an <i>n</i>-fold <i>k</i>-<i>L</i>(<i>p</i>, <i>q</i>)-labeling is called the <i>n</i> -fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling number of <i>G</i> and is denoted by <span>\\(\\lambda _{p,q}^{n}(G)\\)</span>. When <span>\\(p=2, \\)</span> <span>\\(q=1,\\)</span> we use <span>\\(\\lambda ^{n}(G)\\)</span> to replace <span>\\( \\lambda _{2,1}^{n}(G)\\)</span> for simplicity. We study the <i>n</i>-fold <i>L</i>(2, 1) -labeling numbers of Cartesian product of paths and cycles in this paper. We give a necessary and sufficient condition for <span>\\(\\lambda ^{n}(C_{m}\\square P_{2})\\)</span> equals <span>\\(4n+1.\\)</span> Based on this, we determine the exact value of <span>\\( \\lambda ^{2}(C_{m}\\square P_{2})\\)</span> (except for <span>\\(m=5,6\\)</span> and 9) and <span>\\(\\lambda ^{3}(C_{m}\\square P_{2})\\)</span> (except for <span>\\(m=5,6,9,10,13\\)</span> and 17). We also give bounds for <span>\\(\\lambda ^{n}(C_{m}\\square P_{k})\\)</span> when <i>n</i>, <i>m</i> satisfy certain conditions, and from this, we obtain the exact value of <span>\\(\\lambda ^{2}(P_{m}\\square P_{k})\\)</span> (except for the case <span>\\(P_{4}\\square P_{3}\\)</span>).</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"139 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"n-fold L(2, 1)-labelings of Cartesian product of paths and cycles\",\"authors\":\"Fei-Huang Chang, Ma-Lian Chia, Shih-Ang Jiang, David Kuo, Jing-Ho Yan\",\"doi\":\"10.1007/s10878-024-01119-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For two sets of nonnegative integers <i>A</i> and <i>B</i>, the distance between these two sets, denoted by <i>d</i>(<i>A</i>, <i>B</i>), is defined by <span>\\\\(d(A,B)=\\\\min \\\\{|a-b|:a\\\\in A,b\\\\in B\\\\}\\\\)</span>. For a positive integer <i>n</i>, let <span>\\\\(S_{n}\\\\)</span> denote the family <span>\\\\( \\\\{X:X\\\\subseteq {\\\\mathbb {N}} \\\\cup \\\\{0\\\\},|X|=n\\\\}\\\\)</span>. Given a graph <i>G</i> and positive integers <i>n</i>, <i>p</i> and <i>q</i>, an <i>n</i>-fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling of <i>G</i> is a function <span>\\\\(f:V(G)\\\\rightarrow S_{n} \\\\)</span> satisfies <span>\\\\(d(f(u),f(v))\\\\ge p\\\\)</span> if <span>\\\\(d_{G}(u,v)=1\\\\)</span>, and <span>\\\\( d(f(u),f(v))\\\\ge q\\\\)</span> if <span>\\\\(d_{G}(u,v)=2\\\\)</span>. An <i>n</i>-fold <i>k</i>-<i>L</i>(<i>p</i>, <i>q</i>)-labeling <i>f</i> of <i>G</i> is an <i>n</i>-fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling of <i>G</i> with the property that <span>\\\\(\\\\max \\\\{a:a\\\\in \\\\bigcup _{u\\\\in V(G)}f(u)\\\\}\\\\le k\\\\)</span>. The smallest number <i>k</i> to guarantee that <i>G</i> has an <i>n</i>-fold <i>k</i>-<i>L</i>(<i>p</i>, <i>q</i>)-labeling is called the <i>n</i> -fold <i>L</i>(<i>p</i>, <i>q</i>)-labeling number of <i>G</i> and is denoted by <span>\\\\(\\\\lambda _{p,q}^{n}(G)\\\\)</span>. When <span>\\\\(p=2, \\\\)</span> <span>\\\\(q=1,\\\\)</span> we use <span>\\\\(\\\\lambda ^{n}(G)\\\\)</span> to replace <span>\\\\( \\\\lambda _{2,1}^{n}(G)\\\\)</span> for simplicity. We study the <i>n</i>-fold <i>L</i>(2, 1) -labeling numbers of Cartesian product of paths and cycles in this paper. We give a necessary and sufficient condition for <span>\\\\(\\\\lambda ^{n}(C_{m}\\\\square P_{2})\\\\)</span> equals <span>\\\\(4n+1.\\\\)</span> Based on this, we determine the exact value of <span>\\\\( \\\\lambda ^{2}(C_{m}\\\\square P_{2})\\\\)</span> (except for <span>\\\\(m=5,6\\\\)</span> and 9) and <span>\\\\(\\\\lambda ^{3}(C_{m}\\\\square P_{2})\\\\)</span> (except for <span>\\\\(m=5,6,9,10,13\\\\)</span> and 17). We also give bounds for <span>\\\\(\\\\lambda ^{n}(C_{m}\\\\square P_{k})\\\\)</span> when <i>n</i>, <i>m</i> satisfy certain conditions, and from this, we obtain the exact value of <span>\\\\(\\\\lambda ^{2}(P_{m}\\\\square P_{k})\\\\)</span> (except for the case <span>\\\\(P_{4}\\\\square P_{3}\\\\)</span>).</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":\"139 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-04-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-024-01119-9\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-024-01119-9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
n-fold L(2, 1)-labelings of Cartesian product of paths and cycles
For two sets of nonnegative integers A and B, the distance between these two sets, denoted by d(A, B), is defined by \(d(A,B)=\min \{|a-b|:a\in A,b\in B\}\). For a positive integer n, let \(S_{n}\) denote the family \( \{X:X\subseteq {\mathbb {N}} \cup \{0\},|X|=n\}\). Given a graph G and positive integers n, p and q, an n-fold L(p, q)-labeling of G is a function \(f:V(G)\rightarrow S_{n} \) satisfies \(d(f(u),f(v))\ge p\) if \(d_{G}(u,v)=1\), and \( d(f(u),f(v))\ge q\) if \(d_{G}(u,v)=2\). An n-fold k-L(p, q)-labeling f of G is an n-fold L(p, q)-labeling of G with the property that \(\max \{a:a\in \bigcup _{u\in V(G)}f(u)\}\le k\). The smallest number k to guarantee that G has an n-fold k-L(p, q)-labeling is called the n -fold L(p, q)-labeling number of G and is denoted by \(\lambda _{p,q}^{n}(G)\). When \(p=2, \)\(q=1,\) we use \(\lambda ^{n}(G)\) to replace \( \lambda _{2,1}^{n}(G)\) for simplicity. We study the n-fold L(2, 1) -labeling numbers of Cartesian product of paths and cycles in this paper. We give a necessary and sufficient condition for \(\lambda ^{n}(C_{m}\square P_{2})\) equals \(4n+1.\) Based on this, we determine the exact value of \( \lambda ^{2}(C_{m}\square P_{2})\) (except for \(m=5,6\) and 9) and \(\lambda ^{3}(C_{m}\square P_{2})\) (except for \(m=5,6,9,10,13\) and 17). We also give bounds for \(\lambda ^{n}(C_{m}\square P_{k})\) when n, m satisfy certain conditions, and from this, we obtain the exact value of \(\lambda ^{2}(P_{m}\square P_{k})\) (except for the case \(P_{4}\square P_{3}\)).
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.