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Variants of the Gyárfás–Sumner conjecture: Oriented trees and rainbow paths
Gyárfás-Sumner 猜想的变体：定向树和彩虹路径
IF 0.9
3区 数学
Manu Basavaraju, L. Sunil Chandran, Mathew C. Francis, Karthik Murali
{"title":"Variants of the Gyárfás–Sumner conjecture: Oriented trees and rainbow paths","authors":"Manu Basavaraju, L. Sunil Chandran, Mathew C. Francis, Karthik Murali","doi":"10.1002/jgt.23171","DOIUrl":"10.1002/jgt.23171","url":null,"abstract":"<p>Given a finite family <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math> of graphs, we say that a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is “<span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>-free” if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> does not contain any graph in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math> as a subgraph. We abbreviate <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>-free to just “<span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow></math>-free” when <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow></math>. A vertex-colored graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> is called “rainbow” if no two vertices of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> have the same color. Given an integer <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 </mrow></math> and a finite family of graphs <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>, let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> denote the smallest integer such that any properly vertex-colored <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow></math>-free graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> having <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>χ</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>ℓ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"136-161"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199689","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
Corrigendum: The diameter of AT-free graphs
更正：无 AT 图形的直径
IF 0.9
3区 数学
Guillaume Ducoffe
{"title":"Corrigendum: The diameter of AT-free graphs","authors":"Guillaume Ducoffe","doi":"10.1002/jgt.23170","DOIUrl":"10.1002/jgt.23170","url":null,"abstract":"<p>This corrigendum corrects an error found in the proof of correctness of the algorithm by [Ducoffe, JGT, 2022, 99(4), pp. 594–614], Theorem 6. An erroneous result from Deogun and Kratsch was used in the original proof. There are no changes in the algorithm itself.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"848-850"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23170","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199690","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
Removable edges in near-bipartite bricks
近似二方砖中的可移动边缘
IF 0.9
3区 数学
Yipei Zhang, Fuliang Lu, Xiumei Wang, Jinjiang Yuan
{"title":"Removable edges in near-bipartite bricks","authors":"Yipei Zhang, Fuliang Lu, Xiumei Wang, Jinjiang Yuan","doi":"10.1002/jgt.23173","DOIUrl":"10.1002/jgt.23173","url":null,"abstract":"<p>An edge <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow></math> of a matching covered graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is <i>removable</i> if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>e</mi>\u0000 </mrow></math> is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. A nonbipartite matching covered graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a <i>brick</i> if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow></math> and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mover>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>6</mn>\u0000 </msub>\u0000 \u0000 <mo>¯</mo>\u0000 </mover>\u0000 </mrow></math> has at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math> removable edges. A brick <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is <i>near-bipartite</i> if it has a pair of edges <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow></math> such that <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"113-135"},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199691","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
Universal graphs with forbidden wheel minors
具有禁止轮未成年人的通用图
IF 0.9
3区 数学
Thilo Krill
{"title":"Universal graphs with forbidden wheel minors","authors":"Thilo Krill","doi":"10.1002/jgt.23174","DOIUrl":"10.1002/jgt.23174","url":null,"abstract":"<p>Let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>W</mi>\u0000 </mrow></math> be any wheel graph and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> the class of all countable graphs not containing <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>W</mi>\u0000 </mrow></math> as a minor. We show that there exists a graph in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> which contains every graph in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> as an induced subgraph.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"100-112"},"PeriodicalIF":0.9,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23174","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199692","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
On vertex-transitive graphs with a unique hamiltonian cycle
关于具有唯一哈密顿循环的顶点变换图
IF 0.9
3区 数学
Babak Miraftab, Dave Witte Morris
{"title":"On vertex-transitive graphs with a unique hamiltonian cycle","authors":"Babak Miraftab, Dave Witte Morris","doi":"10.1002/jgt.23166","DOIUrl":"10.1002/jgt.23166","url":null,"abstract":"<p>A graph is said to be <i>uniquely hamiltonian</i> if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math> has a Cayley graph of degree <span></span><math>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math> that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math> are outerplanar.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"65-99"},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199693","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
Nonisomorphic two-dimensional algebraically defined graphs over R
R R 上的非同构二维代数定义图
IF 0.9
3区 数学
Brian G. Kronenthal, Joe Miller, Alex Nash, Jacob Roeder, Hani Samamah, Tony W. H. Wong
{"title":"Nonisomorphic two-dimensional algebraically defined graphs over \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23161:jgt23161-math-0001\" wiley:location=\"equation/jgt23161-math-0001.png\"><mrow><mrow><mi mathvariant=\"double-struck\">R</mi></mrow></mrow></math>","authors":"Brian G. Kronenthal, Joe Miller, Alex Nash, Jacob Roeder, Hani Samamah, Tony W. H. Wong","doi":"10.1002/jgt.23161","DOIUrl":"https://doi.org/10.1002/jgt.23161","url":null,"abstract":"<p>For <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>→</mo>\u0000 \u0000 <mi>R</mi>\u0000 </mrow></math>, let <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>f</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> be a two-dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of <span></span><math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow></math> and two vertices <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mi>x</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>x</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow></math> are adjacent if and only if <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <msub>\u0000 <mi>x</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>f</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"50-64"},"PeriodicalIF":0.9,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707817","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
Five-cycle double cover and shortest cycle cover
五周期双覆盖和最短周期覆盖
IF 0.9
3区 数学
Siyan Liu, Rong-Xia Hao, Rong Luo, Cun-Quan Zhang
{"title":"Five-cycle double cover and shortest cycle cover","authors":"Siyan Liu, Rong-Xia Hao, Rong Luo, Cun-Quan Zhang","doi":"10.1002/jgt.23164","DOIUrl":"10.1002/jgt.23164","url":null,"abstract":"<p>The 5-even subgraph cycle double cover conjecture (5-CDC conjecture) asserts that every bridgeless graph has a 5-even subgraph double cover. A shortest even subgraph cover of a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a family of even subgraphs which cover all the edges of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> and the sum of their lengths is minimum. It is conjectured that every bridgeless graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> has an even subgraph cover with total length at most <span></span><math>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>21</mn>\u0000 \u0000 <mn>15</mn>\u0000 </mfrac>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow></math>. In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5-CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> satisfying the sufficient condition has a 4-even subgraph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>-cover with total length at most <span></span><math>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>20</mn>\u0000 \u0000 <mn>15</mn>\u0000 </mfrac>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math>. We also show that every oddness 2 cubic graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> with girth at least 30 has a 5-CDC containing a member of length at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>9</mn>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"39-49"},"PeriodicalIF":0.9,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943300","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
Dense circuit graphs and the planar Turán number of a cycle
密集电路图和循环的平面图兰数
IF 0.9
3区 数学
Ruilin Shi, Zach Walsh, Xingxing Yu
{"title":"Dense circuit graphs and the planar Turán number of a cycle","authors":"Ruilin Shi, Zach Walsh, Xingxing Yu","doi":"10.1002/jgt.23165","DOIUrl":"10.1002/jgt.23165","url":null,"abstract":"<p>The <i>planar Turán number</i> <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>ex</mtext>\u0000 \u0000 <mi>P</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> of a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> is the maximum number of edges in an <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex planar graph without <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> as a subgraph. Let <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow></math> denote the cycle of length <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>. The planar Turán number <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>ex</mtext>\u0000 \u0000 <mi>P</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is known for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>7</mn>\u0000 </mrow></math>. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> so that <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>ex</mtext>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"27-38"},"PeriodicalIF":0.9,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23165","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943337","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
Strong arc decompositions of split digraphs
分裂图的强弧分解
IF 0.9
3区 数学
Jørgen Bang-Jensen, Yun Wang
{"title":"Strong arc decompositions of split digraphs","authors":"Jørgen Bang-Jensen, Yun Wang","doi":"10.1002/jgt.23157","DOIUrl":"10.1002/jgt.23157","url":null,"abstract":"<p>A <i>strong arc decomposition</i> of a digraph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>A</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is a partition of its arc set <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 </mrow></math> into two sets <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow></math> such that the digraph <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is strong for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math>. Bang-Jensen and Yeo conjectured that there is some <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow></math> such that every <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow></math>-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle h","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"5-26"},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943301","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
Stability of Rose Window graphs
玫瑰窗图形的稳定性
IF 0.9
3区 数学
Milad Ahanjideh, István Kovács, Klavdija Kutnar
{"title":"Stability of Rose Window graphs","authors":"Milad Ahanjideh, István Kovács, Klavdija Kutnar","doi":"10.1002/jgt.23162","DOIUrl":"10.1002/jgt.23162","url":null,"abstract":"<p>A graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> is said to be stable if for the direct product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mtext>Aut</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}times {{bf{K}}}_{2},text{Aut}({rm{Gamma }}times {{bf{K}}}_{2})$</annotation>\u0000 </semantics></math> is isomorphic to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Aut</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>Γ</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{Aut}({rm{Gamma }})times {{mathbb{Z}}}_{2}$</annotation>\u0000 </semantics></math>; otherwise, it is called unstable. An unstable graph is called nontrivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all nontrivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"810-832"},"PeriodicalIF":0.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943338","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}
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