{"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}
{"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}
{"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}
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}
{"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":"<p>In matching theory, one of the most fundamental and classical branches of combinatorics, <i>canonical decompositions</i> 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 <i>Dulmage–Mendelsohn</i>, <i>Kotzig–Lovász</i>, and <i>Gallai–Edmonds</i> 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 <i>factor-components</i> 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.</p>","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}
{"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":"<p>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, <span></span><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.</p>","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}
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