Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Paul Wollan
We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small “width”, which is measured in terms of its adhesion and the path-likeness of its torsos.
{"title":"A Grid Theorem for Strong Immersions of Walls","authors":"Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"10.1002/jgt.23245","DOIUrl":"https://doi.org/10.1002/jgt.23245","url":null,"abstract":"<p>We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small “width”, which is measured in terms of its adhesion and the path-likeness of its torsos.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"23-32"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23245","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624421","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}
<div> <p>The recognition of Pfaffian bipartite graphs in polynomial time has been obtained, but this fact is still unknown for Pfaffian nonbipartite graphs. For a path <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </mrow> </semantics></math> and a cycle <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>C</mi> <mi>n</mi> </msub> </mrow> </mrow> </semantics></math>, the Pfaffian graphs of Cartesian products <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mrow> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <msub> <mi>C</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mrow> </semantics></math> were characterized by Lu and Zhang in 2014. Recently, Li and Wang characterized the Pfaffian graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mrow> </semantics></math> for <span></span><math> <semantics> <mrow>
In this paper, we are interested in some problems related to chromatic number and clique number for the class of -free graphs and prove the following the results: (a) If is a connected ()-free graph with , then either is the complement of a bipartite graph or
Ferdinand Ihringer, Paulien Jansen, Linde Lambrecht, Yannick Neyt, Daan Rijpert, Hendrik Van Maldeghem, Magali Victoor
Given a finite Lie incidence geometry, which is either a polar space of rank at least 3 or a strong parapolar space of symplectic rank at least 4 and diameter at most 4, or the parapolar space arising from the line Grassmannian of a projective space of dimension at least 4, we show that its point graph is determined by its local structure. This follows from a more general result, which classifies graphs whose local structure can vary over all local structures of the point graphs of the aforementioned geometries. In particular, this characterises the strongly regular graphs arising from the line Grassmannian of a finite projective space, from the half spin geometry related to the quadric and from the exceptional group of type by their local structure.
{"title":"Local Recognition of the Point Graphs of Some Lie Incidence Geometries","authors":"Ferdinand Ihringer, Paulien Jansen, Linde Lambrecht, Yannick Neyt, Daan Rijpert, Hendrik Van Maldeghem, Magali Victoor","doi":"10.1002/jgt.23243","DOIUrl":"https://doi.org/10.1002/jgt.23243","url":null,"abstract":"<div>\u0000 \u0000 <p>Given a finite Lie incidence geometry, which is either a polar space of rank at least 3 or a strong parapolar space of symplectic rank at least 4 and diameter at most 4, or the parapolar space arising from the line Grassmannian of a projective space of dimension at least 4, we show that its point graph is determined by its local structure. This follows from a more general result, which classifies graphs whose local structure can vary over all local structures of the point graphs of the aforementioned geometries. In particular, this characterises the strongly regular graphs arising from the line Grassmannian of a finite projective space, from the half spin geometry related to the quadric <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>Q</mi>\u0000 \u0000 <mo>+</mo>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>10</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and from the exceptional group of type <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>E</mi>\u0000 \u0000 <mn>6</mn>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> by their local structure.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"518-524"},"PeriodicalIF":0.9,"publicationDate":"2025-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255823","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}
Let be a family of -uniform hypergraphs (henceforth -graphs). An -saturated -graph is a maximal -graph not containing any member of as a subgraph. For integers , let be the collection of all
<div> <p>Let <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> be a graph of order <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>. For an integer <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </mrow> </semantics></math>, a partition <span></span><math> <semantics> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> </semantics></math> of <span></span><math> <semantics> <mrow> <mrow> <mi>V</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> is called a <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>-proper partition of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> if every <span></span><math> <semantics> <mrow> <mrow> <mi>P</mi> <mo>∈</mo> <mi>P</mi> </mrow> </mrow> </semantics></math> induces a <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>-connected subgraph of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>. This concept was introduced by Ferrara et al., and Borozan et al. gave minimum degree conditions for the existence of a <span></span><math> <semantics
Endre Boros, Vladimir Gurvich, Martin Milanič, Yushi Uno
Given a hypergraph , the dual hypergraph of is the hypergraph of all minimal transversals of . The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, and so on. Motivated by a question related to clique transversals in graphs, we study in this paper conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co-NP and that it can be solved in polynomial time for hypergraphs of bounded dimension. For dimension 3, we show that the problem can be reduced to 2-Satisfiability. Our approach has an application in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most , for any fixed .
{"title":"Conformal Hypergraphs: Duality and Implications for the Upper Clique Transversal Problem","authors":"Endre Boros, Vladimir Gurvich, Martin Milanič, Yushi Uno","doi":"10.1002/jgt.23238","DOIUrl":"https://doi.org/10.1002/jgt.23238","url":null,"abstract":"<p>Given a hypergraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℋ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, the dual hypergraph of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℋ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is the hypergraph of all minimal transversals of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℋ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, and so on. Motivated by a question related to clique transversals in graphs, we study in this paper conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in \u0000<span>co-NP</span> and that it can be solved in polynomial time for hypergraphs of bounded dimension. For dimension 3, we show that the problem can be reduced to <span>2-Satisfiability.</span> Our approach has an application in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, for any fixed <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"466-480"},"PeriodicalIF":0.9,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23238","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256575","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}
The total face color polynomial is based upon the Poincaré polynomials of a family of filtered -color homologies. It counts the number of -face colorings of ribbon graphs for each positive integer . As such, it may be seen as a successor of the Penrose polynomial, which at counts 3-edge colorings (and consequently 4-face colorings) of planar trivalent graphs. In this paper, we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.
{"title":"A State Sum for the Total Face Color Polynomial","authors":"Scott Baldridge, Louis H. Kauffman, Ben McCarty","doi":"10.1002/jgt.23239","DOIUrl":"https://doi.org/10.1002/jgt.23239","url":null,"abstract":"<div>\u0000 \u0000 <p>The total face color polynomial is based upon the Poincaré polynomials of a family of filtered <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-color homologies. It counts the number of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-face colorings of ribbon graphs for each positive integer <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. As such, it may be seen as a successor of the Penrose polynomial, which at <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> counts 3-edge colorings (and consequently 4-face colorings) of planar trivalent graphs. In this paper, we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"481-491"},"PeriodicalIF":0.9,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256576","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}
In infinite graph theory, the notion of ends, first introduced by Freudenthal and Jung for locally finite graphs, plays an important role when generalizing statements from finite graphs to infinite ones. Nash-Williams' Tree-Packing Theorem and MacLane's Planarity Criteria are examples of results that allow a topological approach, in which ends may be considered as endpoints of rays. In fact, there are extensive studies in the literature showing that classical (vertex-)connectivity theorems for finite graphs can be discussed regarding ends, in a more general context. However, aiming to generalize results of edge-connectivity, this paper recalls the definition of edge-ends in infinite graphs due to Hahn, Laviolette and Širáň. In terms of that object, we state an edge version of Menger's Theorem (following a previous work of Polat) and generalize the Lovász-Cherkassky Theorem for infinite graphs with edge-ends (inspired by a recent paper of Jacobs, Joó, Knappe, Kurkofka and Melcher).
{"title":"Edge-Connectivity Between Edge-Ends of Infinite Graphs","authors":"Leandro Aurichi, Lucas Real","doi":"10.1002/jgt.23234","DOIUrl":"https://doi.org/10.1002/jgt.23234","url":null,"abstract":"<p>In infinite graph theory, the notion of <i>ends</i>, first introduced by Freudenthal and Jung for locally finite graphs, plays an important role when generalizing statements from finite graphs to infinite ones. Nash-Williams' Tree-Packing Theorem and MacLane's Planarity Criteria are examples of results that allow a topological approach, in which ends may be considered as <i>endpoints</i> of rays. In fact, there are extensive studies in the literature showing that classical (vertex-)connectivity theorems for finite graphs can be discussed regarding ends, in a more general context. However, aiming to generalize results of edge-connectivity, this paper recalls the definition of <i>edge-ends</i> in infinite graphs due to Hahn, Laviolette and Širáň. In terms of that object, we state an edge version of Menger's Theorem (following a previous work of Polat) and generalize the Lovász-Cherkassky Theorem for infinite graphs with edge-ends (inspired by a recent paper of Jacobs, Joó, Knappe, Kurkofka and Melcher).</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"454-465"},"PeriodicalIF":0.9,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23234","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256574","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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