Journal of Graph Theory最新文献

{"title":"A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs","authors":"Yan-Ting Xie,&nbsp;Yong-De Feng,&nbsp;Shou-Jun Xu","doi":"10.1002/jgt.23099","DOIUrl":"10.1002/jgt.23099","url":null,"abstract":"<p>Partial cubes are the graphs which can be embedded into hypercubes. The <i>cube polynomial</i> of a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a counting polynomial of induced hypercubes of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, which is defined as <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow></math>-cubes (hypercubes of dimension <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow></math>) of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. The <i>clique polynomial</i> of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is defined as <span></span><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":null,"pages":null},"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}

{"title":"Flexible list colorings: Maximizing the number of requests satisfied","authors":"Hemanshu Kaul,&nbsp;Rogers Mathew,&nbsp;Jeffrey A. Mudrock,&nbsp;Michael J. Pelsmajer","doi":"10.1002/jgt.23103","DOIUrl":"10.1002/jgt.23103","url":null,"abstract":"<p>Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose <span></span><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>, <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a graph, <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow></math> is a list assignment for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow></math> is a function with nonempty domain <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math> (<span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow></math> is called a request of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow></math>). The triple <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ϵ</mi>\u0000 </mrow></math>-satisfiable if there exists a proper <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Graphs with girth 9 and without longer odd holes are 3-colourable","authors":"Yan Wang,&nbsp;Rong Wu","doi":"10.1002/jgt.23101","DOIUrl":"10.1002/jgt.23101","url":null,"abstract":"<p>For a number <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>l</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math>, let <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow></math>, <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow></math> and <span></span><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 <span></span><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.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Best possible upper bounds on the restrained domination number of cubic graphs","authors":"Boštjan Brešar,&nbsp;Michael A. Henning","doi":"10.1002/jgt.23095","DOIUrl":"10.1002/jgt.23095","url":null,"abstract":"<p>A dominating set in a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a set <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> of vertices such that every vertex in <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math>. A restrained dominating set of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a dominating set <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> with the additional restraint that the graph <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> is isolate-free. The domination number <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. Let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> be a cubic graph of order <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>. A classical result of Reed states that <span></span><math>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Ramsey numbers for multiple copies of sparse graphs","authors":"Aurelio Sulser,&nbsp;Miloš Trujić","doi":"10.1002/jgt.23100","DOIUrl":"10.1002/jgt.23100","url":null,"abstract":"<p>For a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> and an integer <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>, we let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow></math> denote the disjoint union of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math> copies of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math>. In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Bisimplicial separators","authors":"Martin Milanič,&nbsp;Irena Penev,&nbsp;Nevena Pivač,&nbsp;Kristina Vušković","doi":"10.1002/jgt.23098","DOIUrl":"10.1002/jgt.23098","url":null,"abstract":"<p>A <i>minimal separator</i> of a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a set <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> separates <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow></math> from <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow></math> in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, but no proper subset of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> does. For an integer <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-<i>simplicial</i> if it can be covered by <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math> cliques and denote by <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-simplicial. We show that for each <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}

{"title":"Edge-minimum saturated \u0000 \u0000 k\u0000 -planar drawings","authors":"Steven Chaplick,&nbsp;Fabian Klute,&nbsp;Irene Parada,&nbsp;Jonathan Rollin,&nbsp;Torsten Ueckerdt","doi":"10.1002/jgt.23097","DOIUrl":"10.1002/jgt.23097","url":null,"abstract":"<p>For a class <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> of drawings of loopless (multi-)graphs in the plane, a drawing <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math> is <i>saturated</i> when the addition of any edge to <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> results in <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math> times, and the classes <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> of all <span></span><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 <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex saturated <span></span><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 <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex saturated <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings have <span></span><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":null,"pages":null},"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}

{"title":"The chromatic number of heptagraphs","authors":"Di Wu,&nbsp;Baogang Xu,&nbsp;Yian Xu","doi":"10.1002/jgt.23094","DOIUrl":"10.1002/jgt.23094","url":null,"abstract":"<p>A pentagraph is a graph without cycles of length 3 or 4 and without induced cycles of odd length at least 7, and a heptagraph is one without cycles of length less than 7 and without induced cycles of odd length at least 9. Chudnovsky and Seymour proved that every pentagraph is 3-colorable. In this paper, we show that every heptagraph is 3-colorable.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181763","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}

{"title":"Circular flows in mono-directed signed graphs","authors":"Jiaao Li,&nbsp;Reza Naserasr,&nbsp;Zhouningxin Wang,&nbsp;Xuding Zhu","doi":"10.1002/jgt.23092","DOIUrl":"10.1002/jgt.23092","url":null,"abstract":"<p>In this paper, the concept of circular <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> is an orientation on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> and <span></span><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 <span></span><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":null,"pages":null},"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}

{"title":"Ramsey numbers upon vertex deletion","authors":"Yuval Wigderson","doi":"10.1002/jgt.23093","DOIUrl":"10.1002/jgt.23093","url":null,"abstract":"<p>Given a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, its Ramsey number <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation> $N$</annotation>\u0000 </semantics></math> so that every two-coloring of <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"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}