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The maximum number of odd cycles in a planar graph
IF 0.9
3区 数学
Emily Heath, Ryan R. Martin, Chris Wells
{"title":"The maximum number of odd cycles in a planar graph","authors":"Emily Heath, Ryan R. Martin, Chris Wells","doi":"10.1002/jgt.23197","DOIUrl":"https://doi.org/10.1002/jgt.23197","url":null,"abstract":"<p>How many copies of a fixed odd cycle, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{2m+1}$</annotation>\u0000 </semantics></math>, can a planar graph contain? We answer this question asymptotically for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $min {2,3,4}$</annotation>\u0000 </semantics></math> and prove a bound which is tight up to a factor of 3/2 for all other values of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>. This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation> $mu $</annotation>\u0000 </semantics></math> on the edges of some clique maximizes the probability that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> edges sampled independently from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation> $mu $</annotation>\u0000 </semantics></math> form either a cycle or a path?</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"745-780"},"PeriodicalIF":0.9,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23197","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143456010","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
Odd chromatic number of graph classes
IF 0.9
3区 数学
Rémy Belmonte, Ararat Harutyunyan, Noleen Köhler, Nikolaos Melissinos
{"title":"Odd chromatic number of graph classes","authors":"Rémy Belmonte, Ararat Harutyunyan, Noleen Köhler, Nikolaos Melissinos","doi":"10.1002/jgt.23200","DOIUrl":"https://doi.org/10.1002/jgt.23200","url":null,"abstract":"<p>A graph is called <i>odd</i> (respectively, <i>even</i>) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an <i>odd colouring</i> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. Scott proved that a connected graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-odd colourable if it can be partitioned into at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> odd induced subgraphs. The <i>odd chromatic number of</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, denoted by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>χ</mi>\u0000 \u0000 <mtext>odd</mtext>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${chi }_{text{odd}}(G)$</annotation>\u0000 </semantics></math>, is the minimum integer <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> for which <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-odd colourable. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. We fir","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"722-744"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23200","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The effect of symmetry-preserving operations on 3-connectivity
IF 0.9
3区 数学
Heidi Van den Camp
{"title":"The effect of symmetry-preserving operations on 3-connectivity","authors":"Heidi Van den Camp","doi":"10.1002/jgt.23196","DOIUrl":"https://doi.org/10.1002/jgt.23196","url":null,"abstract":"<p>In 2017, Brinkmann, Goetschalckx and Schein introduced a very general way of describing operations on embedded graphs that preserve all orientation-preserving symmetries of the graph. This description includes all well-known operations such as Dual, Truncation and Ambo. As these operations are applied locally, they are called local orientation-preserving symmetry-preserving operations (lopsp-operations). In this text, we will use the general description of these operations to determine their effect on 3-connectivity. Recently it was proved that all lopsp-operations preserve 3-connectivity of graphs that have face-width at least three. We present a simple condition that characterises exactly which lopsp-operations preserve 3-connectivity for all embedded graphs, even for those with face-width less than three.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"672-704"},"PeriodicalIF":0.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455666","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
Edge-arc-disjoint paths in semicomplete mixed graphs
IF 0.9
3区 数学
J. Bang-Jensen, Y. Wang
{"title":"Edge-arc-disjoint paths in semicomplete mixed graphs","authors":"J. Bang-Jensen, Y. Wang","doi":"10.1002/jgt.23199","DOIUrl":"https://doi.org/10.1002/jgt.23199","url":null,"abstract":"<p>The so-called <i>weak-2-linkage problem</i> asks for a given digraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>=</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 <mo>,</mo>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=(V,A)$</annotation>\u0000 </semantics></math> and distinct vertices <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>t</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>t</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${s}_{1},{s}_{2},{t}_{1},{t}_{2}$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> whether <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> has arc-disjoint paths <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${P}_{1},{P}_{2}$</annotation>\u0000 </semantics></math> so that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${P}_{i}$</annotation>\u0000 </semantics></math> is an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>s</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"705-721"},"PeriodicalIF":0.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23199","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455667","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
Graphs with girth 2 ℓ + 1 $2ell +1$ and without longer odd holes are 3-colorable
IF 0.9
3区 数学
Rong Chen
{"title":"Graphs with girth \u0000 \u0000 \u0000 2\u0000 ℓ\u0000 +\u0000 1\u0000 \u0000 $2ell +1$\u0000 and without longer odd holes are 3-colorable","authors":"Rong Chen","doi":"10.1002/jgt.23195","DOIUrl":"https://doi.org/10.1002/jgt.23195","url":null,"abstract":"<p>For a number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $ell ge 2$</annotation>\u0000 </semantics></math>, let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>ℓ</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{G}}}_{ell }$</annotation>\u0000 </semantics></math> denote the family of graphs which have girth <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>ℓ</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $2ell +1$</annotation>\u0000 </semantics></math> and have no odd hole with length greater than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>ℓ</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $2ell +1$</annotation>\u0000 </semantics></math>. Wu et al. conjectured that every graph in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mo>⋃</mo>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msub>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>ℓ</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${bigcup }_{ell ge 2}{{mathscr{G}}}_{ell }$</annotation>\u0000 </semantics></math> is 3-colorable. Chudnovsky et al. and Wu et al., respectively, proved that every graph in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{G}}}_{2}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{G}}}_{3}$</annotation>\u0000 </semantics></math> is 3-colorable. In this paper, we prove that every graph in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mo>⋃</mo>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 <mo>≥</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 </msub>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"661-671"},"PeriodicalIF":0.9,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455776","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
Face-simple minimal quadrangulations of surfaces
IF 0.9
3区 数学
Sarah Abusaif, Warren Singh, Timothy Sun
{"title":"Face-simple minimal quadrangulations of surfaces","authors":"Sarah Abusaif, Warren Singh, Timothy Sun","doi":"10.1002/jgt.23198","DOIUrl":"https://doi.org/10.1002/jgt.23198","url":null,"abstract":"<p>For each surface besides the sphere, projective plane, and Klein bottle, we construct a face-simple minimal quadrangulation, that is, a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our result answers a question of Liu, Ellingham, and Ye while providing a simpler proof of their main result. The inductive construction is based on an earlier idea for finding near-quadrangular embeddings of the complete graphs using the diamond sum operation.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"647-655"},"PeriodicalIF":0.9,"publicationDate":"2024-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120103","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
Improved bounds on the cop number when forbidding a minor
IF 0.9
3区 数学
Franklin Kenter, Erin Meger, Jérémie Turcotte
{"title":"Improved bounds on the cop number when forbidding a minor","authors":"Franklin Kenter, Erin Meger, Jérémie Turcotte","doi":"10.1002/jgt.23194","DOIUrl":"https://doi.org/10.1002/jgt.23194","url":null,"abstract":"<p>Andreae proved that the cop number of connected <span></span><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:jgt23194:jgt23194-math-0001\" wiley:location=\"equation/jgt23194-math-0001.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-minor-free graphs is bounded for every graph <span></span><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:jgt23194:jgt23194-math-0002\" wiley:location=\"equation/jgt23194-math-0002.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>. In particular, the cop number is at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>h</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0003\" wiley:location=\"equation/jgt23194-math-0003.png\"><mrow><mrow><mo>unicode{x02223}</mo><mi>E</mi><mrow><mo>(</mo><mrow><mi>H</mi><mo>unicode{x02212}</mo><mi>h</mi></mrow><mo>)</mo></mrow><mo>unicode{x02223}</mo></mrow></mrow></math></annotation>\u0000 </semantics></math> if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 \u0000 <mo>−</mo>\u0000 \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:jgt23194:jgt23194-math-0004\" wiley:location=\"equation/jgt23194-math-0004.png\"><mrow><mrow>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"620-646"},"PeriodicalIF":0.9,"publicationDate":"2024-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143110557","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
A localized approach for Turán number of long cycles
IF 0.9
3区 数学
Kai Zhao, Xiao-Dong Zhang
{"title":"A localized approach for Turán number of long cycles","authors":"Kai Zhao, Xiao-Dong Zhang","doi":"10.1002/jgt.23191","DOIUrl":"https://doi.org/10.1002/jgt.23191","url":null,"abstract":"<p>Let <span></span><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:jgt23191:jgt23191-math-0001\" wiley:location=\"equation/jgt23191-math-0001.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> be a simple graph and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>c</mi>\u0000 \u0000 <mi>G</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>e</mi>\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:jgt23191:jgt23191-math-0002\" wiley:location=\"equation/jgt23191-math-0002.png\"><mrow><mrow><msub><mi>c</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\u0000 </semantics></math> denote the length of the longest cycle containing an edge <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23191:jgt23191-math-0003\" wiley:location=\"equation/jgt23191-math-0003.png\"><mrow><mrow><mi>e</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> if there exists a cycle containing <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23191:jgt23191-math-0004\" wiley:location=\"equation/jgt23191-math-0004.png\"><mrow><mrow><mi>e</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>, and 2 otherwise. We prove that the summation of 2 divided by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>c</mi>\u0000 \u0000 <mi>G</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"582-607"},"PeriodicalIF":0.9,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113620","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
Towards Nash-Williams orientation conjecture for infinite graphs
IF 0.9
3区 数学
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":"<p>In 1960 Nash-Williams proved that an edge-connectivity of <span></span><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 <span></span><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 <i>lifting graph</i>, proved that <span></span><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 <span></span><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 <span></span><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
Switchover phenomenon for general graphs
IF 0.9
3区 数学
Dániel Keliger, László Lovász, Tamás Ferenc Móri, Gergely Ódor
{"title":"Switchover phenomenon for general graphs","authors":"Dániel Keliger, László Lovász, Tamás Ferenc Móri, Gergely Ódor","doi":"10.1002/jgt.23184","DOIUrl":"https://doi.org/10.1002/jgt.23184","url":null,"abstract":"<p>We study SIR-type epidemics (susceptible-infected-resistant) on graphs in two scenarios: (i) when the initial infections start from a well-connected central region and (ii) when initial infections are distributed uniformly. Previously, Ódor et al. demonstrated on a few random graph models that the expectation of the total number of infections undergoes a switchover phenomenon; the central region is more dangerous for small infection rates, while for large rates, the uniform seeding is expected to infect more nodes. We rigorously prove this claim under mild, deterministic assumptions on the underlying graph. If we further assume that the central region has a large enough expansion, the second moment of the degree distribution is bounded and the number of initial infections is comparable to the number of vertices, the difference between the two scenarios is shown to be macroscopic.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"560-581"},"PeriodicalIF":0.9,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23184","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113536","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
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