Journal of Graph Theory最新文献

{"title":"Treewidth Versus Clique Number. V. Further Connections With Tree-Independence Number","authors":"Claire Hilaire, Martin Milanič, Ðorđe Vasić","doi":"10.1002/jgt.70036","DOIUrl":"https://doi.org/10.1002/jgt.70036","url":null,"abstract":"<p>We continue the study of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>tw</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ω</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-bounded graph classes, that is, hereditary graph classes in which large treewidth is witnessed by the presence of a large clique, and the relation of this property to boundedness of the tree-independence number, a graph parameter introduced independently by Yolov in 2018 and by Dallard, Milanič, and Štorgel in 2024. Dallard et al. showed that bounded tree-independence number is sufficient for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>tw</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ω</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-boundedness, and conjectured that the converse holds. While this conjecture has been recently disproved, it is still interesting to determine classes where the conjecture holds; for example, the conjecture is still open for graph classes excluding an induced star, as well as for finitely many forbidden induced subgraphs. In this paper, we identify further families of graph classes where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>tw</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ω</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-boundedness is equivalent to bounded tree-independence number. We settle a number of cases of finitely many forbidden induced subgraphs, obtain several equivalent characterizations of <span></span><ma","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 3","pages":"337-351"},"PeriodicalIF":1.0,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70036","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148069247","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":"Identifying Codes in Triangle-Free Graphs of Bounded Maximum Degree","authors":"Dipayan Chakraborty, Florent Foucaud, Michael A. Henning, Tuomo Lehtilä","doi":"10.1002/jgt.70025","DOIUrl":"https://doi.org/10.1002/jgt.70025","url":null,"abstract":"<div>\u0000 \u0000 <p>An <i>identifying code</i> of a closed-twin-free graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> such that any two vertices in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> have a distinct, nonempty intersection between their closed neighborhood and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. It was conjectured that there exists a constant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> such that for every connected closed-twin-free graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and maximum degree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, the graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> admits an identifying code of size at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfenced>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 3","pages":"306-336"},"PeriodicalIF":1.0,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148098871","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":"There Are Finitely Many 5-Vertex-Critical \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 \u0000 P\u0000 6\u0000 \u0000 ,\u0000 \u0000 \u0000 bull\u0000 \u0000 \u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 -Free Graphs","authors":"Yiao Ju, Jorik Jooken, Jan Goedgebeur, Shenwei Huang","doi":"10.1002/jgt.70027","DOIUrl":"https://doi.org/10.1002/jgt.70027","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we are interested in 4-colouring algorithms for graphs that do not contain an induced path on six vertices nor an induced bull, that is, the graph with vertex set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and edge set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 3","pages":"255-266"},"PeriodicalIF":1.0,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148091351","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":"Cubic Graphs With Edges in Exactly One Perfect Matching","authors":"Jan Goedgebeur, Davide Mattiolo, Giuseppe Mazzuoccolo, Jarne Renders, Isaak H. Wolf","doi":"10.1002/jgt.70024","DOIUrl":"https://doi.org/10.1002/jgt.70024","url":null,"abstract":"<div>\u0000 \u0000 <p>Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. In fact, it is known that for a very large fraction of bridgeless cubic graphs, every edge belongs to at least two distinct perfect matchings. In this paper, we study the class of non-double covered cubic graphs, that is, graphs having an edge, called lonely edge, which belongs to exactly one perfect matching. First of all, we provide a reduction of the problem to the subclass <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>U</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of 3-connected cubic graphs. Then, we furnish an inductive characterization of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>U</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and we study properties related to the count of lonely edges. In particular, denoting by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>U</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> the subclass of graphs of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>U</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> lonely edges, we prove that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>U</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is empty for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mn>6</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and we present a complete characterization for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 3","pages":"276-289"},"PeriodicalIF":1.0,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148099752","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":"On Delta-Wye Reducibility of Toroidal Graphs","authors":"Isidoro Gitler,&nbsp;Gustavo Sandoval–Angeles","doi":"10.1002/jgt.70028","DOIUrl":"https://doi.org/10.1002/jgt.70028","url":null,"abstract":"<div>\u0000 \u0000 <p>In 2006, Yu proved that there are at least 68 billion forbidden minors for delta-wye reducibility. It is an open problem to prove that this is the complete list of forbidden minors. We prove that only eight graphs from Yu's list admit an embedding in the torus. These graphs are the Petersen family of graphs together with the graph obtained from the complete bipartite graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <mn>5</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>5</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> by deleting a perfect matching. Moreover, we show that graphs that admit a toroidal embedding of representativity greater than three contain one of these eight toroidal graphs as a minor. Hence, they are not delta-wye reducible. In contrast, we prove that any graph that admits a toroidal embedding with representativity less than two is delta-wye reducible; in fact, 1-terminal reducible. We also prove that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> admits a toroidal embedding without disjoint essential cycles, then <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is delta-wye reducible. Finally, we show that if a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> has a toroidal embedding that is a 3-connected triangulation of the torus, then <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is not delta-wye reducible.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 3","pages":"183-197"},"PeriodicalIF":1.0,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148085509","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":"Sensitivity and Hamming Graphs","authors":"Sara Asensio,&nbsp;Yuval Filmus,&nbsp;Ignacio García-Marco,&nbsp;Kolja Knauer","doi":"10.1002/jgt.70034","DOIUrl":"https://doi.org/10.1002/jgt.70034","url":null,"abstract":"<p>For any <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> we show that the Hamming graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>m</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> admits an imbalanced partition into <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-ary Sensitivity Conjecture of Asensio, García-Marco, and Knauer. On the other hand, we prove their weaker <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-ary Sensitivity Conjecture by showing that the sensitivity of any <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-ary function is bounded from below by a polynomial expression in its degree.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 3","pages":"296-305"},"PeriodicalIF":1.0,"publicationDate":"2026-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148099751","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":"A Strong Structural Stability of \u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 2\u0000 k\u0000 +\u0000 1\u0000 \u0000 \u0000 \u0000 \u0000 -Free Graphs","authors":"Zilong Yan,&nbsp;Yuejian Peng","doi":"10.1002/jgt.70022","DOIUrl":"https://doi.org/10.1002/jgt.70022","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Füredi and Gunderson showed that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is achieved only on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⌊&lt;/mo&gt;\u0000 \u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 \u0000 &lt;mo&gt;⌋&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⌈&lt;/mo&gt;\u0000 \u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 \u0000 &lt;mo&gt;⌉&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 2","pages":"151-160"},"PeriodicalIF":1.0,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147686268","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":"Sharp Exponents for Bipartite Erdős-Rado Numbers","authors":"Dániel Dobák,&nbsp;Eion Mulrenin","doi":"10.1002/jgt.70016","DOIUrl":"https://doi.org/10.1002/jgt.70016","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;The Erdős–Rado canonization theorem generalizes Ramsey's theorem to edge-colorings with an unbounded number of colors, in the sense that for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; sufficiently large, any edge-coloring of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; will yield some copy of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; which is colored according to one of four canonical patterns. In this paper, we show that in the bipartite setting, the bipartite Erdős–Rado number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfies\u0000\u0000 &lt;/p&gt;&lt;div&gt;&lt;span&gt;&lt;span&gt;&lt;!--FIGURE--&gt;&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;log&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 2","pages":"96-102"},"PeriodicalIF":1.0,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147686961","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":"Spectral Radius and Edge-Disjoint Spanning Trees of Graphs With Prescribed Edge Connectivity","authors":"Yuke Zhang,&nbsp;Dandan Fan","doi":"10.1002/jgt.70017","DOIUrl":"https://doi.org/10.1002/jgt.70017","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;τ&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denote the edge connectivity and the maximum number of edge-disjoint spanning trees contained in a graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, respectively. Fan, Gu, and Lin studied a tight spectral condition of a connected graph to guarantee &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;τ&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The famous theorem of Tutte and Nash-Williams implies that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;τ&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 2","pages":"128-144"},"PeriodicalIF":1.0,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147686269","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":"Properly Colored Cycles in Edge-Colored Balanced Bipartite Graphs","authors":"Tingting Han,&nbsp;Hajo Broersma,&nbsp;Shenggui Zhang,&nbsp;Yandong Bai","doi":"10.1002/jgt.70008","DOIUrl":"https://doi.org/10.1002/jgt.70008","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denote a (not necessarily properly) edge-colored balanced bipartite graph on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices, that is, in which every edge is assigned a color. A cycle &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is called properly colored if any two consecutive edges of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; have distinct colors. A properly colored cycle-factor of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 2","pages":"37-53"},"PeriodicalIF":1.0,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147686124","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}