Journal of Graph Theory最新文献

{"title":"Tree Independence Number III. Thetas, Prisms and Stars","authors":"Maria Chudnovsky,&nbsp;Sepehr Hajebi,&nbsp;Nicolas Trotignon","doi":"10.1002/jgt.70088","DOIUrl":"https://doi.org/10.1002/jgt.70088","url":null,"abstract":"<p>We prove that for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>N</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> there exists <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>τ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>τ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>t</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>N</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> such that every (theta, prism, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>t</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>)-free graph has tree independence number at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>τ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> (where we allow “prisms” to have one path of length zero).</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 4","pages":"619-633"},"PeriodicalIF":1.0,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70088","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149177481","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":"Stable Cuts, NAC-Colourings and Flexible Realisations of Graphs","authors":"Katie Clinch,&nbsp;Dániel Garamvölgyi,&nbsp;John Haslegrave,&nbsp;Tony Huynh,&nbsp;Jan Legerský,&nbsp;Anthony Nixon","doi":"10.1002/jgt.70064","DOIUrl":"https://doi.org/10.1002/jgt.70064","url":null,"abstract":"&lt;p&gt;A (2-dimensional) realisation of a graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a pair &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; maps the vertices of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. We say that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is rigid if every generic realisation of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is rigid; otherwise, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is flexible. In this paper, we investigate the relationship between stable cuts and graphs which are either flexible or admit a flexible (not necessarily generic) realisation with positive edge lengths. We strengthen a result of Chen and Yu, who proved that every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-vertex graph with at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"411-427"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70064","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148903700","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":"Semistrong Edge Coloring and \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 0\u0000 ,\u0000 1\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 -Relaxed Strong Edge Coloring of Graphs","authors":"Yuquan Lin,&nbsp;Wensong Lin","doi":"10.1002/jgt.70074","DOIUrl":"https://doi.org/10.1002/jgt.70074","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In this work, we study two relaxations of the well-known strong edge coloring. A &lt;i&gt;semistrong edge coloring&lt;/i&gt; of 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; is an edge coloring in which every color class forms a matching &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;M&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that every edge 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;M&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is incident with (at least) one vertex of degree 1 in the subgraph 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;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; induced by the vertices covered by &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;M&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. For any two nonnegative integers &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;s&lt;/mi&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;t&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;t&lt;/mi&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;-&lt;i&gt;relaxed strong edge coloring&lt;/i&gt; 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;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an edge coloring in which, for every edge &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 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"490-507"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148903801","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":"Tight Minimum Degree Conditions for Apex-Outerplanar Minors and Subdivisions in Graphs and Digraphs","authors":"Chun-Hung Liu,&nbsp;Youngho Yoo","doi":"10.1002/jgt.70073","DOIUrl":"https://doi.org/10.1002/jgt.70073","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Motivated by Hadwiger's conjecture and related problems for list-coloring, we study graphs &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;H&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for which every graph with minimum degree at least &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;H&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;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; contains &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;H&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as a minor. We prove that a large class of apex-outerplanar graphs satisfies this property. Our result gives the first examples of such graphs whose vertex cover numbers are significantly larger than half the number of its vertices, which breaks a barrier for attacking related coloring problems via extremal functions, and recovers all known such graphs that have arbitrarily large maximum degree. Our proof can be adapted to directed graphs to show that 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;mover&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the digraph obtained from a directed cycle or an in-arborescence by adding an apex source, then every digraph with minimum out-degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mover&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"470-489"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148904030","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":"Inducibility of Four-Vertex Tournaments","authors":"Dalton Burke,&nbsp;Bernard Lidický,&nbsp;Florian Pfender,&nbsp;Michael Phillips","doi":"10.1002/jgt.70075","DOIUrl":"https://doi.org/10.1002/jgt.70075","url":null,"abstract":"<div>\u0000 \u0000 <p>We determine the inducibility of all tournaments with at most four vertices together with the extremal constructions. The four-vertex tournament containing an oriented <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and one source vertex has a particularly interesting extremal construction, first conjectured by Bożyk, Grzesik, and Kielak. It is an unbalanced blow-up of an edge, where the sink vertex is replaced by a quasi-random tournament and the source vertex is iteratively replaced by a copy of the construction itself.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"508-522"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148903774","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":"A Min–Max Relation on Dicuts and Dijoins in Weighted Chordal Digraphs","authors":"Gérard Cornuéjols,&nbsp;Siyue Liu,&nbsp;R. Ravi","doi":"10.1002/jgt.70076","DOIUrl":"https://doi.org/10.1002/jgt.70076","url":null,"abstract":"<p>In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the maximum size of a packing of dijoins. This has been disproved. However, the unweighted version conjectured by Woodall remains open. We prove that the Edmonds–Giles conjecture is true if the underlying undirected graph is chordal. We also give a strongly polynomial-time algorithm to construct such a packing.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"462-469"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70076","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148904029","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":"Erratum to “Proper Edge Colorings of Planar Graphs With Rainbow C4-s”","authors":"","doi":"10.1002/jgt.70122","DOIUrl":"https://doi.org/10.1002/jgt.70122","url":null,"abstract":"&lt;p&gt;In the proof of the following theorem (Theorem 1.4(i) in [&lt;span&gt;1&lt;/span&gt;]) a small error was made, the purpose of this note is to present the corrected proof. Indeed, when induction was applied on the number of edges for the graph &lt;i&gt;H&lt;/i&gt; obtained from &lt;i&gt;G&lt;/i&gt; by removing the edge {&lt;i&gt;v, w&lt;/i&gt;}, it is possible that an edge from &lt;i&gt;v&lt;/i&gt; gets the same color as an edge from &lt;i&gt;w&lt;/i&gt;. Then when we add back the edge {&lt;i&gt;v, w&lt;/i&gt;}, this might create a forbidden four-cycle. In the corrected proof we apply induction on the number of vertices instead.&lt;/p&gt;&lt;p&gt;\u0000 &lt;/p&gt;&lt;p&gt;\u0000 &lt;/p&gt;&lt;p&gt;If &lt;i&gt;w&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt; has degree zero in &lt;i&gt;H&lt;/i&gt;[&lt;i&gt;W&lt;/i&gt;] then the colors on edges within &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;w&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are not in a four-cycle containing {&lt;i&gt;v, w&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;}, thus in (b) no edges need to be forbidden, (d) contributes ∆ + 2, altogether at most ∆ − 1 + (&lt;i&gt;i&lt;/i&gt; − 1) + ∆ + 2 = 2∆ + &lt;i&gt;i&lt;/i&gt; colors are forbidden on {&lt;i&gt;v, w&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;}.&lt;/p&gt;&lt;p&gt;If &lt;i&gt;w&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt; has degree one in &lt;i&gt;H&lt;/i&gt;[&lt;i&gt;W&lt;/i&gt;] then at most three edges within &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;w&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are in a four-cycle containing {&lt;i&gt;v, w&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;}, thus (b) contributes at most 3 and (d) contributes at most ∆ + 1 (because &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;w&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;). Altogether at","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"538-539"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70122","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148904317","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":"Ban-Linial's Conjecture and Halin Snarks","authors":"Xiaoyu Li,&nbsp;Rong-Xia Hao,&nbsp;Yilun Luo,&nbsp;Siyan Liu","doi":"10.1002/jgt.70080","DOIUrl":"https://doi.org/10.1002/jgt.70080","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;A 2-bisection is a 2-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;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (not necessarily proper) such that each color class has the same cardinality, and each monochromatic component has at most 2 vertices. It was conjectured by Ban and Linial that every cubic graph admits a 2-bisection, except for the Petersen graph. After discovering some examples, Esperet et al. [J. Graph Theory, 2017, 86(2), 149–158] further verified a conjecture for the family of bridgeless cubic graph that admits a 2-bisection, except for the Petersen graph. Zerafa [J. Graph Theory, 2022, 101(3), 511–520] proved that each treelike snark admits a 2-bisection. It is known that the family of Halin snarks &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;H&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;;&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;D&lt;/mi&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 a superset of treelike snarks. In the same paper, Zerafa raised a problem that what assumption needed to be made with regard to the building block &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;D&lt;/mi&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; of a Halin snark for the verification of the 2-bisection conjecture. This problem is solved in this paper and it is proved that each Halin snark admits a 2-bisection if the multipole &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;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the building blocks has an all-2-balanced and an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"527-537"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148903783","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":"A Short Proof of Tuza's Conjecture for Weak Saturation in Hypergraphs","authors":"Nikolai Terekhov","doi":"10.1002/jgt.70082","DOIUrl":"https://doi.org/10.1002/jgt.70082","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Given an &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;r&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-uniform hypergraph &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;H&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and a positive integer &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 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the weak saturation 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;mtext&gt;wsat&lt;/mtext&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;mi&gt;H&lt;/mi&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 the minimum number of edges in an &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;r&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-uniform hypergraph &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;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; 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;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices such that the missing edges 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;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; can be added, one at a time, so that each added edge creates a 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;mi&gt;H&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Shapira and Tyomkyn (Proceedings of the American Mathematical Society, 2023) proved Tuza's conjecture on the asymptotic behaviour 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;mtext&gt;wsa","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"523-526"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148903873","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":"Linear Versus Centred Colouring via Pseudogrids","authors":"Prosenjit Bose,&nbsp;Vida Dujmović,&nbsp;Hussein Houdrouge,&nbsp;Mehrnoosh Javarsineh,&nbsp;Pat Morin","doi":"10.1002/jgt.70072","DOIUrl":"https://doi.org/10.1002/jgt.70072","url":null,"abstract":"&lt;p&gt;A &lt;i&gt;centred colouring&lt;/i&gt; of a graph is a vertex colouring in which every connected subgraph contains a vertex whose colour is unique and a &lt;i&gt;linear colouring&lt;/i&gt; is a vertex colouring in which every (not-necessarily induced) path contains a vertex whose colour is unique. For 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;, the &lt;i&gt;centred chromatic number&lt;/i&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;msub&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mtext&gt;cen&lt;/mtext&gt;\u0000 &lt;/msub&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 the &lt;i&gt;linear chromatic number&lt;/i&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;msub&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mtext&gt;lin&lt;/mtext&gt;\u0000 &lt;/msub&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 minimum number of distinct colours required for a centred, respectively, linear colouring 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;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. From these definitions, it follows immediately 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;msub&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mtext&gt;lin&lt;/mtext&gt;\u0000 &lt;/msub&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 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 3","pages":"448-461"},"PeriodicalIF":1.0,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70072","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148904028","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}