Journal of Graph Theory最新文献_第8页

A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs 部分立方体的立方多项式与其交叉图的小块多项式之间的关系

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-04-23 DOI: 10.1002/jgt.23099

Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu

{"title":"A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs","authors":"Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu","doi":"10.1002/jgt.23099","DOIUrl":"10.1002/jgt.23099","url":null,"abstract":"Partial cubes are the graphs which can be embedded into hypercubes. The cube polynomial of a graph <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a counting polynomial of induced hypercubes of <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, which is defined as <math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≔</mo>\u0000 \u0000 <msub>\u0000 <mo>∑</mo>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>⩾</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>α</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <msup>\u0000 <mi>x</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msup>\u0000 </mrow></math>, where <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>α</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is the number of induced <math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow></math>-cubes (hypercubes of dimension <math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow></math>) of <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. The clique polynomial of <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is defined as <math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 \u0000 <mi>l</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"907-922"},"PeriodicalIF":0.9,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Flexible list colorings: Maximizing the number of requests satisfied 灵活的列表着色：最大限度地满足请求数量

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-04-18 DOI: 10.1002/jgt.23103

Hemanshu Kaul, Rogers Mathew, Jeffrey A. Mudrock, Michael J. Pelsmajer

{"title":"Flexible list colorings: Maximizing the number of requests satisfied","authors":"Hemanshu Kaul, Rogers Mathew, Jeffrey A. Mudrock, Michael J. Pelsmajer","doi":"10.1002/jgt.23103","DOIUrl":"10.1002/jgt.23103","url":null,"abstract":"Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose <math>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>ϵ</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>, <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a graph, <math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow></math> is a list assignment for <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, and <math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow></math> is a function with nonempty domain <math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> such that <math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>L</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> for each <math>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math> (<math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow></math> is called a request of <math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow></math>). The triple <math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>L</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>r</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is <math>\u0000 \u0000 <mrow>\u0000 <mi>ϵ</mi>\u0000 </mrow></math>-satisfiable if there exists a proper <math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"887-906"},"PeriodicalIF":0.9,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140627463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Graphs with girth 9 and without longer odd holes are 3-colourable 周长为 9 且没有较长奇数孔的图形是 3 可取的

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-04-11 DOI: 10.1002/jgt.23101

Yan Wang, Rong Wu

{"title":"Graphs with girth 9 and without longer odd holes are 3-colourable","authors":"Yan Wang, Rong Wu","doi":"10.1002/jgt.23101","DOIUrl":"10.1002/jgt.23101","url":null,"abstract":"For a number <math>\u0000 \u0000 <mrow>\u0000 <mi>l</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math>, let <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>l</mi>\u0000 </msub>\u0000 </mrow></math> denote the family of graphs which have girth <math>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>l</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math> and have no odd hole with length greater than <math>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>l</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>. Wu, Xu and Xu conjectured that every graph in <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mo>⋃</mo>\u0000 \u0000 <mrow>\u0000 <mi>l</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>l</mi>\u0000 </msub>\u0000 </mrow></math> is 3-colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow></math>, <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow></math> and <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mo>⋃</mo>\u0000 \u0000 <mrow>\u0000 <mi>l</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>5</mn>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>l</mi>\u0000 </msub>\u0000 </mrow></math> is 3-colourable, respectively. In this paper, we prove that every graph in <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow></math> is 3-colourable. This confirms Wu, Xu and Xu's conjecture.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"871-886"},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Best possible upper bounds on the restrained domination number of cubic graphs 立方图的约束支配数的最佳上限

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-04-11 DOI: 10.1002/jgt.23095

Boštjan Brešar, Michael A. Henning

{"title":"Best possible upper bounds on the restrained domination number of cubic graphs","authors":"Boštjan Brešar, Michael A. Henning","doi":"10.1002/jgt.23095","DOIUrl":"10.1002/jgt.23095","url":null,"abstract":"A dominating set in a graph <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a set <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> of vertices such that every vertex in <math>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>⧹</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow></math> is adjacent to a vertex in <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math>. A restrained dominating set of <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a dominating set <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> with the additional restraint that the graph <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow></math> obtained by removing all vertices in <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> is isolate-free. The domination number <math>\u0000 \u0000 <mrow>\u0000 <mi>γ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> and the restrained domination number <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>γ</mi>\u0000 \u0000 <mi>r</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. Let <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> be a cubic graph of order <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>. A classical result of Reed states that <math>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"763-815"},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23095","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Ramsey numbers for multiple copies of sparse graphs 稀疏图形多副本的拉姆齐数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-04-11 DOI: 10.1002/jgt.23100

Aurelio Sulser, Miloš Trujić

{"title":"Ramsey numbers for multiple copies of sparse graphs","authors":"Aurelio Sulser, Miloš Trujić","doi":"10.1002/jgt.23100","DOIUrl":"10.1002/jgt.23100","url":null,"abstract":"For a graph <math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> and an integer <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>, we let <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow></math> denote the disjoint union of <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math> copies of <math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math>. In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow></math>, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant <math>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>c</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> such that <math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>α</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>c</mi>\u0000 </mrow></math>, provided <math>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"843-870"},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23100","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Bisimplicial separators 二等分隔符

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-04-11 DOI: 10.1002/jgt.23098

Martin Milanič, Irena Penev, Nevena Pivač, Kristina Vušković

{"title":"Bisimplicial separators","authors":"Martin Milanič, Irena Penev, Nevena Pivač, Kristina Vušković","doi":"10.1002/jgt.23098","DOIUrl":"10.1002/jgt.23098","url":null,"abstract":"A minimal separator of a graph <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a set <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> such that there exist vertices <math>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>⧹</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow></math> with the property that <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> separates <math>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow></math> from <math>\u0000 \u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow></math> in <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, but no proper subset of <math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> does. For an integer <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow></math>, we say that a minimal separator is <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-simplicial if it can be covered by <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math> cliques and denote by <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow></math> the class of all graphs in which each minimal separator is <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-simplicial. We show that for each <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"816-842"},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23098","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Edge-minimum saturated k -planar drawings 边缘最小饱和 k 平面绘图

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-03-28 DOI: 10.1002/jgt.23097

Steven Chaplick, Fabian Klute, Irene Parada, Jonathan Rollin, Torsten Ueckerdt

{"title":"Edge-minimum saturated \u0000 \u0000 k\u0000 -planar drawings","authors":"Steven Chaplick, Fabian Klute, Irene Parada, Jonathan Rollin, Torsten Ueckerdt","doi":"10.1002/jgt.23097","DOIUrl":"10.1002/jgt.23097","url":null,"abstract":"For a class <math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> of drawings of loopless (multi-)graphs in the plane, a drawing <math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math> is saturated when the addition of any edge to <math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> results in <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>D</mi>\u0000 \u0000 <mo>′</mo>\u0000 </msup>\u0000 \u0000 <mo>∉</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math>—this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math> times, and the classes <math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> of all <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex saturated <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex saturated <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings have <math>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>2</mn>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"741-762"},"PeriodicalIF":0.9,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23097","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The chromatic number of heptagraphs 七段体的色度数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-03-19 DOI: 10.1002/jgt.23094

Di Wu, Baogang Xu, Yian Xu

引用次数: 0

Circular flows in mono-directed signed graphs 单向有符号图中的循环流动

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-03-19 DOI: 10.1002/jgt.23092

Jiaao Li, Reza Naserasr, Zhouningxin Wang, Xuding Zhu

{"title":"Circular flows in mono-directed signed graphs","authors":"Jiaao Li, Reza Naserasr, Zhouningxin Wang, Xuding Zhu","doi":"10.1002/jgt.23092","DOIUrl":"10.1002/jgt.23092","url":null,"abstract":"In this paper, the concept of circular <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-flow in a mono-directed signed graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>,</mo>\u0000 <mi>σ</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(G,sigma )$</annotation>\u0000 </semantics></math> is introduced. That is a pair <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>,</mo>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(D,f)$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> is an orientation on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>→</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>r</mi>\u0000 <mo>,</mo>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $f:E(G)to (-r,r)$</annotation>\u0000 </semantics></math> satisfies that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>e</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∣</mo>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>r</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"686-710"},"PeriodicalIF":0.9,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Ramsey numbers upon vertex deletion 顶点删除后的拉姆齐数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-03-18 DOI: 10.1002/jgt.23093

Yuval Wigderson

{"title":"Ramsey numbers upon vertex deletion","authors":"Yuval Wigderson","doi":"10.1002/jgt.23093","DOIUrl":"10.1002/jgt.23093","url":null,"abstract":"Given a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, its Ramsey number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $r(G)$</annotation>\u0000 </semantics></math> is the minimum <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation> $N$</annotation>\u0000 </semantics></math> so that every two-coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>N</mi>\u0000 </msub>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $E({K}_{N})$</annotation>\u0000 </semantics></math> contains a monochromatic copy of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation> ${{G}_{n}}$</annotation>\u0000 </semantics></math> so that in any Ramsey coloring for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"663-675"},"PeriodicalIF":0.9,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23093","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0