{"title":"某些平面图形的 L(3,2,1)- 标记","authors":"","doi":"10.1016/j.tcs.2024.114881","DOIUrl":null,"url":null,"abstract":"<div><div>Given a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> of maximum degree <em>Δ</em>, denoting by <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> the distance in <em>G</em> between nodes <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>V</mi></math></span>, an <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling of <em>G</em> is an assignment <em>l</em> from <em>V</em> to the set of non-negative integers such that <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>3</mn></math></span> if <em>x</em> and <em>y</em> are adjacent, <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>2</mn></math></span> if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>, and <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>3</mn></math></span>, for all <em>x</em> and <em>y</em> in <em>V</em>. The <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the smallest positive integer such that <em>G</em> admits an <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling with labels from <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>}</mo></math></span>.</div><div>In this paper, the <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number of certain planar graphs is determined, proving that it is linear in <em>Δ</em>, although the general upper bound for the <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number of planar graphs is quadratic in <em>Δ</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"L(3,2,1)-labeling of certain planar graphs\",\"authors\":\"\",\"doi\":\"10.1016/j.tcs.2024.114881\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Given a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> of maximum degree <em>Δ</em>, denoting by <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> the distance in <em>G</em> between nodes <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>V</mi></math></span>, an <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling of <em>G</em> is an assignment <em>l</em> from <em>V</em> to the set of non-negative integers such that <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>3</mn></math></span> if <em>x</em> and <em>y</em> are adjacent, <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>2</mn></math></span> if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>, and <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>3</mn></math></span>, for all <em>x</em> and <em>y</em> in <em>V</em>. The <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the smallest positive integer such that <em>G</em> admits an <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling with labels from <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>}</mo></math></span>.</div><div>In this paper, the <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number of certain planar graphs is determined, proving that it is linear in <em>Δ</em>, although the general upper bound for the <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number of planar graphs is quadratic in <em>Δ</em>.</div></div>\",\"PeriodicalId\":49438,\"journal\":{\"name\":\"Theoretical Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-09-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304397524004985\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524004985","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
给定一个最大度数为 Δ 的图 G=(V,E) ,用 d(x,y) 表示 G 中节点 x,y∈V 之间的距离,L(3,2、1)-labeling is an assignment l from V to the set of non-negative integers such that |l(x)-l(y)|≥3 if x and y are adjacent, |l(x)-l(y)|≥2 if d(x,y)=2, and |l(x)-l(y)|≥1 if d(x,y)=3, for all x and y in V.本文确定了某些平面图的 L(3,2,1)数,证明它在Δ中是线性的,尽管平面图的 L(3,2,1)数的一般上限在Δ中是二次方。
Given a graph of maximum degree Δ, denoting by the distance in G between nodes , an -labeling of G is an assignment l from V to the set of non-negative integers such that if x and y are adjacent, if , and if , for all x and y in V. The -number is the smallest positive integer such that G admits an -labeling with labels from .
In this paper, the -number of certain planar graphs is determined, proving that it is linear in Δ, although the general upper bound for the -number of planar graphs is quadratic in Δ.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.