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On a Norine–Thomas conjecture concerning minimal bricks
关于最小砖块的诺林-托马斯猜想
IF 0.9
3区 数学
Xing Feng
{"title":"On a Norine–Thomas conjecture concerning minimal bricks","authors":"Xing Feng","doi":"10.1002/jgt.23175","DOIUrl":"10.1002/jgt.23175","url":null,"abstract":"<p>A 3-connected graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a <i>brick</i> if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow></math> has a perfect matching, for each pair <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> of vertices of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. A brick <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is <i>minimal</i> if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>e</mi>\u0000 </mrow></math> ceases to be a brick for every edge <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\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 </mrow></math>. Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow></math> such that every minimal brick <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> has at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>α</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 </mrow></math> cubic vertices. In this paper, we prove this conjecture holds for all minimal bricks of an average degree no less than 23/5. As its corollary, we show that each minimal brick on <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math> vertices contains more than <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>5</mn>\u0000 </mrow></math> vertices of degree at most four.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"162-172"},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199688","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
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}
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