Journal of Graph Theory最新文献_第5页

Towards Nash-Williams orientation conjecture for infinite graphs 无限图的Nash-Williams方向猜想

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-08 DOI: 10.1002/jgt.23192

Amena Assem

{"title":"Towards Nash-Williams orientation conjecture for infinite graphs","authors":"Amena Assem","doi":"10.1002/jgt.23192","DOIUrl":"https://doi.org/10.1002/jgt.23192","url":null,"abstract":"In 1960 Nash-Williams proved that an edge-connectivity of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0001\" wiley:location=\"equation/jgt23192-math-0001.png\"><mrow><mrow><mn>2</mn><mi>k</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> is sufficient for a finite graph to have a <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0002\" wiley:location=\"equation/jgt23192-math-0002.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>8</mn>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0003\" wiley:location=\"equation/jgt23192-math-0003.png\"><mrow><mrow><mn>8</mn><mi>k</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-edge-connected infinite graphs admit a <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0004\" wiley:location=\"equation/jgt23192-math-0004.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-arc-connected orientation. Here we improve this result for the class of one-ended locally finite graphs and show that an edge-connectivity of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"608-619"},"PeriodicalIF":0.9,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23192","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113544","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

Basilica: New canonical decomposition in matching theory Basilica：匹配理论中的新正则分解

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-07 DOI: 10.1002/jgt.23190

Nanao Kita

{"title":"Basilica: New canonical decomposition in matching theory","authors":"Nanao Kita","doi":"10.1002/jgt.23190","DOIUrl":"https://doi.org/10.1002/jgt.23190","url":null,"abstract":"In matching theory, one of the most fundamental and classical branches of combinatorics, canonical decompositions of graphs are powerful and versatile tools that form the basis of this theory. However, the abilities of the known canonical decompositions, that is, the Dulmage–Mendelsohn, Kotzig–Lovász, and Gallai–Edmonds decompositions, are limited because they are only applicable to particular classes of graphs, such as bipartite graphs, or they are too sparse to provide sufficient information. To overcome these limitations, we introduce a new canonical decomposition that is applicable to all graphs and provides much finer information. This decomposition also provides the answer to the longstanding absence of a canonical decomposition that is nontrivially applicable to general graphs with perfect matchings. We focus on the notion of factor-components as the fundamental building blocks of a graph; through the factor-components, our new canonical decomposition states how a graph is organized and how it contains all the maximum matchings. The main results that constitute our new theory are the following: (i) a canonical partial order over the set of factor-components, which describes how a graph is constructed from its factor-components; (ii) a generalization of the Kotzig–Lovász decomposition, which shows the inner structure of each factor-component in the context of the entire graph; and (iii) a canonically described interrelationship between (i) and (ii), which integrates these two results into a unified theory of a canonical decomposition. These results are obtained in a self-contained way, and our proof of the generalized Kotzig–Lovász decomposition contains a shortened and self-contained proof of the classical counterpart.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"508-542"},"PeriodicalIF":0.9,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23190","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113128","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 graphs for which large books are Ramsey good 关于图表，拉姆齐擅长写大部头的书

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-07 DOI: 10.1002/jgt.23193

Meng Liu, Yusheng Li

{"title":"On graphs for which large books are Ramsey good","authors":"Meng Liu, Yusheng Li","doi":"10.1002/jgt.23193","DOIUrl":"https://doi.org/10.1002/jgt.23193","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0001\" wiley:location=\"equation/jgt23193-math-0001.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> be a graph and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0002\" wiley:location=\"equation/jgt23193-math-0002.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> a connected graph. Then <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0003\" wiley:location=\"equation/jgt23193-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> is said to be <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0004\" wiley:location=\"equation/jgt23193-math-0004.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-good if the Ramsey number <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0005\" wiley:location=\"equation/jgt23193-math-0005.png\"><mrow><mrow&gt","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"543-559"},"PeriodicalIF":0.9,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113129","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

The complexity of the perfect matching-cut problem 完美匹配割问题的复杂性

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-06 DOI: 10.1002/jgt.23167

Valentin Bouquet, Christophe Picouleau

{"title":"The complexity of the perfect matching-cut problem","authors":"Valentin Bouquet, Christophe Picouleau","doi":"10.1002/jgt.23167","DOIUrl":"https://doi.org/10.1002/jgt.23167","url":null,"abstract":"PERFECT MATCHING-CUT is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three, and for bipartite graphs of diameter four. We show that there exist polynomial-time algorithms for the following classes of graphs: claw-free, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>P</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23167:jgt23167-math-0001\" wiley:location=\"equation/jgt23167-math-0001.png\"><mrow><mrow><msub><mi>P</mi><mn>5</mn></msub></mrow></mrow></math></annotation>\u0000 </semantics></math>-free, diameter two, bipartite with diameter three, and graphs with bounded treewidth.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"432-462"},"PeriodicalIF":0.9,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23167","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113000","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 a conjecture that strengthens Kundu's k

k

-factor theorem 关于强化Kundu k因子定理的一个猜想

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-06 DOI: 10.1002/jgt.23177

James M. Shook

{"title":"On a conjecture that strengthens Kundu's \u0000 \u0000 \u0000 \u0000 k\u0000 \u0000 \u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0001\" wiley:location=\"equation/jgt23177-math-0001.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math>\u0000 -factor theorem","authors":"James M. Shook","doi":"10.1002/jgt.23177","DOIUrl":"https://doi.org/10.1002/jgt.23177","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>π</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>d</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>d</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0002\" wiley:location=\"equation/jgt23177-math-0002.png\"><mrow><mrow><mi>unicode{x003C0}</mi><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>,</mo><mo>unicode{x02026}</mo><mo>,</mo><msub><mi>d</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\u0000 </semantics></math> be a nonincreasing degree sequence with even <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0003\" wiley:location=\"equation/jgt23177-math-0003.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>. In 1974, Kundu showed that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>π</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"463-491"},"PeriodicalIF":0.9,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112839","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

On the Constructor–Blocker game 在建造者-拦阻者游戏中

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-06 DOI: 10.1002/jgt.23186

József Balogh, Ce Chen, Sean English

{"title":"On the Constructor–Blocker game","authors":"József Balogh, Ce Chen, Sean English","doi":"10.1002/jgt.23186","DOIUrl":"https://doi.org/10.1002/jgt.23186","url":null,"abstract":"In the Constructor–Blocker game, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0001\" wiley:location=\"equation/jgt23186-math-0001.png\"><mrow><mrow><msub><mi>K</mi><mi>n</mi></msub></mrow></mrow></math></annotation>\u0000 </semantics></math>. For given graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0002\" wiley:location=\"equation/jgt23186-math-0002.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0003\" wiley:location=\"equation/jgt23186-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>, Constructor can only claim edges that leave her graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0004\" wiley:location=\"equation/jgt23186-math-0004.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-free, while Blocker has no restrictions. Constructor's goal is to build as many copies of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0005\" wiley:location=\"equation/jgt23186-math-0005.png\"><mrow><mrow><mi>H</mi></mrow></mrow></","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"492-507"},"PeriodicalIF":0.9,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112840","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

Super stable tensegrities and the Colin de Verdière number ν

unicode{x003BD}

超稳定张拉整体和柯林常数ν unicode{x003BD}

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-02 DOI: 10.1002/jgt.23188

Ryoshun Oba, Shin-ichi Tanigawa

{"title":"Super stable tensegrities and the Colin de Verdière number \u0000 \u0000 \u0000 \u0000 ν\u0000 \u0000 \u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0001\" wiley:location=\"equation/jgt23188-math-0001.png\"><mrow><mrow><mi>unicode{x003BD}</mi></mrow></mrow></math>","authors":"Ryoshun Oba, Shin-ichi Tanigawa","doi":"10.1002/jgt.23188","DOIUrl":"https://doi.org/10.1002/jgt.23188","url":null,"abstract":"A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars and struts connected by cables with tension. We introduce the super stability number of a multigraph as the maximum dimension that a multigraph can be realized as a super stable tensegrity, and show that it equals the Colin de Verdière number <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ν</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0002\" wiley:location=\"equation/jgt23188-math-0002.png\"><mrow><mrow><mi>unicode{x003BD}</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> minus one. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as three-dimensional super stable tensegrities. We also show that, for any fixed <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0003\" wiley:location=\"equation/jgt23188-math-0003.png\"><mrow><mrow><mi>d</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>, there is an infinite family of 3-regular graphs that can be realized as <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23188:jgt23188-math-0004\" wiley:location=\"equation/jgt23188-math-0004.png\"><mrow><mrow><mi>d</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-dimensional injective super stable tensegrities.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"401-431"},"PeriodicalIF":0.9,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23188","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143110740","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

Counting circuit double covers 计数电路双盖

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-10-02 DOI: 10.1002/jgt.23187

Radek Hušek, Robert Šámal

{"title":"Counting circuit double covers","authors":"Radek Hušek, Robert Šámal","doi":"10.1002/jgt.23187","DOIUrl":"https://doi.org/10.1002/jgt.23187","url":null,"abstract":"We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${C}_{k}$</annotation>\u0000 </semantics></math> for some <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${2}^{n/2-1}$</annotation>\u0000 </semantics></math> circuit double covers and we show an infinite class of graphs for which this bound is tight.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"374-395"},"PeriodicalIF":0.9,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23187","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868031","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

Non-Hamiltonian Cartesian products of two even dicycles 两个偶双环的非哈密顿笛卡尔积

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-30 DOI: 10.1002/jgt.23185

Kenta Noguchi, Carol T. Zamfirescu

{"title":"Non-Hamiltonian Cartesian products of two even dicycles","authors":"Kenta Noguchi, Carol T. Zamfirescu","doi":"10.1002/jgt.23185","DOIUrl":"https://doi.org/10.1002/jgt.23185","url":null,"abstract":"In this note it is proven that there exist infinitely many positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $a$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $b$</annotation>\u0000 </semantics></math> such that the Cartesian product of a directed cycle of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $2a$</annotation>\u0000 </semantics></math> and a directed cycle of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $2b$</annotation>\u0000 </semantics></math> is non-Hamiltonian. In particular, the Cartesian product of an 880-dicycle and a 4368-dicycle is non-Hamiltonian. We also prove that there is no such graph on fewer than <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>880</mn>\u0000 \u0000 <mo>⋅</mo>\u0000 \u0000 <mn>4368</mn>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>843</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>840</mn>\u0000 </mrow>\u0000 <annotation> $880cdot 4368=3,843,840$</annotation>\u0000 </semantics></math> vertices, which is rather astonishing.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"368-373"},"PeriodicalIF":0.9,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869244","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

An extension of Nash-Williams and Tutte's Theorem 纳什-威廉姆斯和图特定理的推广

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-09-26 DOI: 10.1002/jgt.23189

Xuqian Fang, Daqing Yang

{"title":"An extension of Nash-Williams and Tutte's Theorem","authors":"Xuqian Fang, Daqing Yang","doi":"10.1002/jgt.23189","DOIUrl":"https://doi.org/10.1002/jgt.23189","url":null,"abstract":"The celebrated Nash-Williams and Tutte's Theorem states that a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> edge-disjoint spanning trees if and only if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${nu }_{f}(G)ge k$</annotation>\u0000 </semantics></math>, where\u0000\u0000 In this paper, we prove that, for integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $kge 0$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $dge 1$</annotation>\u0000 </semantics></math>, if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mfrac>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"361-367"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869216","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