Pub Date : 2025-08-08DOI: 10.1016/j.jctb.2025.07.006
Marcin Briański, Daniel Král', Ander Lamaison
Contraction⁎-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph G is the minimum height of a rooted forest whose closure contains G by proving the following for every matroid M (except the trivial case when M consists of loops and coloops only): the contraction⁎-depth of M plus one is equal to the minimum contraction-depth of a matroid containing M as a restriction.
{"title":"Closure property of contraction-depth of matroids","authors":"Marcin Briański, Daniel Král', Ander Lamaison","doi":"10.1016/j.jctb.2025.07.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2025.07.006","url":null,"abstract":"Contraction<ce:sup loc=\"post\">⁎</ce:sup>-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph <ce:italic>G</ce:italic> is the minimum height of a rooted forest whose closure contains <ce:italic>G</ce:italic> by proving the following for every matroid <ce:italic>M</ce:italic> (except the trivial case when <ce:italic>M</ce:italic> consists of loops and coloops only): the contraction<ce:sup loc=\"post\">⁎</ce:sup>-depth of <ce:italic>M</ce:italic> plus one is equal to the minimum contraction-depth of a matroid containing <ce:italic>M</ce:italic> as a restriction.","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"53 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given any ε>0 we prove that every sufficiently large n-vertex 3-graph H where every pair of vertices is contained in at least (1/3+ε)n edges contains a copy of C10, i.e. the tight cycle on 10 vertices. In fact we obtain the same conclusion for every cycle Cℓ with ℓ≥19.
{"title":"The codegree Turán density of 3-uniform tight cycles","authors":"Simón Piga, Nicolás Sanhueza-Matamala, Mathias Schacht","doi":"10.1016/j.jctb.2025.07.007","DOIUrl":"https://doi.org/10.1016/j.jctb.2025.07.007","url":null,"abstract":"Given any <mml:math altimg=\"si1.svg\"><mml:mi>ε</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:math> we prove that every sufficiently large <ce:italic>n</ce:italic>-vertex 3-graph <ce:italic>H</ce:italic> where every pair of vertices is contained in at least <mml:math altimg=\"si2.svg\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:mo linebreak=\"badbreak\" linebreakstyle=\"after\">+</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>n</mml:mi></mml:math> edges contains a copy of <mml:math altimg=\"si3.svg\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub></mml:math>, i.e. the tight cycle on 10 vertices. In fact we obtain the same conclusion for every cycle <mml:math altimg=\"si36.svg\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:math> with <mml:math altimg=\"si5.svg\"><mml:mi>ℓ</mml:mi><mml:mo>≥</mml:mo><mml:mn>19</mml:mn></mml:math>.","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"27 1","pages":"1-6"},"PeriodicalIF":1.4,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144901792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-04DOI: 10.1016/j.jctb.2025.07.004
Bo Ning , Long-Tu Yuan
In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least k, then it contains a cycle of length at least except for some special families of graphs. Our results imply several previous classical theorems including a deep and old result by Voss. We point out our result on stability in Bondy's theorem can directly imply a positive solution (in a slight stronger form) to the following problem: Is there a polynomial time algorithm to decide whether a 2-connected graph G on n vertices has a cycle of length at least ? This problem originally motivates the recent study on algorithmic aspects of Dirac's theorem by Fomin, Golovach, Sagunov, and Simonov, although a stronger problem was solved by them by completely different methods. Our theorem can also help us to determine all extremal graphs for wheels on odd number of vertices. We also discuss the relationship between our results and some previous problems and theorems in spectral graph theory and generalized Turán problems.
{"title":"Stability in Bondy's theorem on paths and cycles","authors":"Bo Ning , Long-Tu Yuan","doi":"10.1016/j.jctb.2025.07.004","DOIUrl":"10.1016/j.jctb.2025.07.004","url":null,"abstract":"<div><div>In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least <em>k</em>, then it contains a cycle of length at least <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></math></span> except for some special families of graphs. Our results imply several previous classical theorems including a deep and old result by Voss. We point out our result on stability in Bondy's theorem can directly imply a positive solution (in a slight stronger form) to the following problem: Is there a polynomial time algorithm to decide whether a 2-connected graph <em>G</em> on <em>n</em> vertices has a cycle of length at least <span><math><mi>min</mi><mo></mo><mo>{</mo><mn>2</mn><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>}</mo></math></span>? This problem originally motivates the recent study on algorithmic aspects of Dirac's theorem by Fomin, Golovach, Sagunov, and Simonov, although a stronger problem was solved by them by completely different methods. Our theorem can also help us to determine all extremal graphs for wheels on odd number of vertices. We also discuss the relationship between our results and some previous problems and theorems in spectral graph theory and generalized Turán problems.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 213-239"},"PeriodicalIF":1.2,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-31DOI: 10.1016/j.jctb.2025.07.005
Debsoumya Chakraborti , Tuan Tran
Fix and consider a multipartite graph G with maximum degree at most , parts of the same size n, and where every vertex has at most neighbors in any part . Loh and Sudakov proved that any such G has an independent transversal. They further conjectured that the vertex set of G can be decomposed into pairwise disjoint independent transversals. In the present paper, we resolve this conjecture approximately by showing that G contains pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.
{"title":"Approximate packing of independent transversals in locally sparse graphs","authors":"Debsoumya Chakraborti , Tuan Tran","doi":"10.1016/j.jctb.2025.07.005","DOIUrl":"10.1016/j.jctb.2025.07.005","url":null,"abstract":"<div><div>Fix <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> and consider a multipartite graph <em>G</em> with maximum degree at most <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span>, parts <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of the same size <em>n</em>, and where every vertex has at most <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> neighbors in any part <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Loh and Sudakov proved that any such <em>G</em> has an independent transversal. They further conjectured that the vertex set of <em>G</em> can be decomposed into pairwise disjoint independent transversals. In the present paper, we resolve this conjecture approximately by showing that <em>G</em> contains <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 187-212"},"PeriodicalIF":1.2,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-18DOI: 10.1016/j.jctb.2025.07.003
Huan Zhou, Jialu Zhu, Xuding Zhu
A graph G is called degree-truncated k-choosable if for every list assignment L with for each vertex v, G is L-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated 16-choosable). We further prove that for an arbitrary proper minor closed family of graphs, let s be the minimum integer such that for some t, then there is a constant k such that every s-connected graph other than a GDP tree is degree-truncated DP-k-colourable (and hence degree-truncated k-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface Σ, there is a constant k such that every 3-connected non-complete graph embeddable on Σ is degree-truncated DP-k-colourable (and hence degree-truncated k-choosable). The s-connectedness for graphs in (and 3-connectedness for graphs embeddable on Σ) is necessary, as for any positive integer k, ( is planar) is not degree-truncated k-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.
{"title":"Degree-truncated choosability of graphs","authors":"Huan Zhou, Jialu Zhu, Xuding Zhu","doi":"10.1016/j.jctb.2025.07.003","DOIUrl":"10.1016/j.jctb.2025.07.003","url":null,"abstract":"<div><div>A graph <em>G</em> is called degree-truncated <em>k</em>-choosable if for every list assignment <em>L</em> with <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mi>min</mi><mo></mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mi>k</mi><mo>}</mo></math></span> for each vertex <em>v</em>, <em>G</em> is <em>L</em>-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated 16-choosable). We further prove that for an arbitrary proper minor closed family <span><math><mi>G</mi></math></span> of graphs, let <em>s</em> be the minimum integer such that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>∉</mo><mi>G</mi></math></span> for some <em>t</em>, then there is a constant <em>k</em> such that every <em>s</em>-connected graph <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span> other than a GDP tree is degree-truncated DP-<em>k</em>-colourable (and hence degree-truncated <em>k</em>-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface Σ, there is a constant <em>k</em> such that every 3-connected non-complete graph embeddable on Σ is degree-truncated DP-<em>k</em>-colourable (and hence degree-truncated <em>k</em>-choosable). The <em>s</em>-connectedness for graphs in <span><math><mi>G</mi></math></span> (and 3-connectedness for graphs embeddable on Σ) is necessary, as for any positive integer <em>k</em>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msub><mo>∈</mo><mi>G</mi></math></span> (<span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> is planar) is not degree-truncated <em>k</em>-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 171-186"},"PeriodicalIF":1.2,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-18DOI: 10.1016/j.jctb.2025.07.001
Michael Krivelevich , Alan Lew , Peleg Michaeli
A graph is called d-rigid if there exists a generic embedding of its vertex set into such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal d such that the graph is d-rigid. We present new sufficient conditions for the d-rigidity of a graph in terms of the existence of “rigid partitions”—partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz.
As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random -regular graphs are typically d-rigid, demonstrate the existence of a giant d-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting disjoint connected dominating sets is d-rigid. This implies a weak version of the Lovász–Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for d-rigidity in the random graph process typically coincides with the hitting time for minimum degree d.
{"title":"Rigid partitions: From high connectivity to random graphs","authors":"Michael Krivelevich , Alan Lew , Peleg Michaeli","doi":"10.1016/j.jctb.2025.07.001","DOIUrl":"10.1016/j.jctb.2025.07.001","url":null,"abstract":"<div><div>A graph is called <em>d</em>-rigid if there exists a generic embedding of its vertex set into <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal <em>d</em> such that the graph is <em>d</em>-rigid. We present new sufficient conditions for the <em>d</em>-rigidity of a graph in terms of the existence of “rigid partitions”—partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz.</div><div>As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random <span><math><mi>C</mi><mi>d</mi><mi>log</mi><mo></mo><mi>d</mi></math></span>-regular graphs are typically <em>d</em>-rigid, demonstrate the existence of a giant <em>d</em>-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span> disjoint connected dominating sets is <em>d</em>-rigid. This implies a weak version of the Lovász–Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for <em>d</em>-rigidity in the random graph process typically coincides with the hitting time for minimum degree <em>d</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 126-170"},"PeriodicalIF":1.2,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-17DOI: 10.1016/j.jctb.2025.07.002
Anton Bernshteyn , Abhishek Dhawan
Vizing's theorem states that every graph G of maximum degree Δ can be properly edge-colored using colors. The fastest currently known -edge-coloring algorithm for general graphs is due to Sinnamon and runs in time , where and . We investigate the case when Δ is constant, i.e., . In this regime, the runtime of Sinnamon's algorithm is , which can be improved to , as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only , which is obviously best possible. Prior to this work, no linear-time -edge-coloring algorithm was known for any . Using some of the same ideas, we also develop new algorithms for -edge-coloring in the model of distributed computation. Namely, when Δ is constant, we design a deterministic algorithm with running time and a randomized algorithm with running time . Although our focus is on the constant Δ regime, our results remain interesting for Δ up to , since the dependence of their running time on Δ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.
{"title":"Fast algorithms for Vizing's theorem on bounded degree graphs","authors":"Anton Bernshteyn , Abhishek Dhawan","doi":"10.1016/j.jctb.2025.07.002","DOIUrl":"10.1016/j.jctb.2025.07.002","url":null,"abstract":"<div><div>Vizing's theorem states that every graph <em>G</em> of maximum degree Δ can be properly edge-colored using <span><math><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> colors. The fastest currently known <span><math><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time <span><math><mi>O</mi><mo>(</mo><mi>m</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span>, where <span><math><mi>n</mi><mo>≔</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> and <span><math><mi>m</mi><mo>≔</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span>. We investigate the case when Δ is constant, i.e., <span><math><mi>Δ</mi><mo>=</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. In this regime, the runtime of Sinnamon's algorithm is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>, which can be improved to <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, which is obviously best possible. Prior to this work, no linear-time <span><math><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-edge-coloring algorithm was known for any <span><math><mi>Δ</mi><mo>⩾</mo><mn>4</mn></math></span>. Using some of the same ideas, we also develop new algorithms for <span><math><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-edge-coloring in the <span><math><mi>LOCAL</mi></math></span> model of distributed computation. Namely, when Δ is constant, we design a deterministic <span><math><mi>LOCAL</mi></math></span> algorithm with running time <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>5</mn></mrow></msup><mo></mo><mi>n</mi><mo>)</mo></math></span> and a randomized <span><math><mi>LOCAL</mi></math></span> algorithm with running time <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>n</mi><mo>)</mo></math></span>. Although our focus is on the constant Δ regime, our results remain interesting for Δ up to <span><math><msup><mrow><mi>log</mi></mrow><mrow><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo></mo><mi>n</mi></math></span>, since the dependence of their running time on Δ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 69-125"},"PeriodicalIF":1.2,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-27DOI: 10.1016/j.jctb.2025.06.003
Nathan Bowler, Florian Reich
In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids.
In this paper we develop a universal end space theory based on connectoids: the end spaces of connectoids unify the existing end spaces of undirected and directed graphs and establish end spaces for bidirected graphs, hypergraphs and finitary matroids.
The main result shows that the tangle-like description of ends in undirected graphs, called directions, extends to connectoids: there is a one-to-one correspondence between the “directions” of a connectoid and its ends. Furthermore, we generalise normal trees of undirected graphs to connectoids and show that normal trees represent the ends of a connectoid as they do for undirected graphs.
{"title":"Connectoids I: A universal end space theory","authors":"Nathan Bowler, Florian Reich","doi":"10.1016/j.jctb.2025.06.003","DOIUrl":"10.1016/j.jctb.2025.06.003","url":null,"abstract":"<div><div>In this series we introduce and investigate the concept of <em>connectoids</em>, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids.</div><div>In this paper we develop a universal end space theory based on connectoids: the end spaces of connectoids unify the existing end spaces of undirected and directed graphs and establish end spaces for bidirected graphs, hypergraphs and finitary matroids.</div><div>The main result shows that the tangle-like description of ends in undirected graphs, called <em>directions</em>, extends to connectoids: there is a one-to-one correspondence between the “directions” of a connectoid and its ends. Furthermore, we generalise normal trees of undirected graphs to connectoids and show that normal trees represent the ends of a connectoid as they do for undirected graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 584-606"},"PeriodicalIF":1.2,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144515440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-20DOI: 10.1016/j.jctb.2025.06.002
Luke Postle , Evelyne Smith-Roberge
We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas and by Dvořák and Kawarabayashi, respectively. Using results on hyperbolic families due to Postle and Thomas, we show that this implies that there exists a universal constant c such that if Σ is a surface of Euler genus g, every graph of edge-width at least embedded in Σ is 5-correspondence colourable. This is asymptotically best possible, and improves upon the best known edge-width bound due to Kim, Kostochka, Li, and Zhu. Using results of Dvořák and Kawarabayashi, we show further that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.
{"title":"Hyperbolicity theorems for correspondence colouring","authors":"Luke Postle , Evelyne Smith-Roberge","doi":"10.1016/j.jctb.2025.06.002","DOIUrl":"10.1016/j.jctb.2025.06.002","url":null,"abstract":"<div><div>We generalize a framework of list colouring results to <em>correspondence colouring</em>. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form <em>hyperbolic families</em>. Analogous results for list colouring were shown by Postle and Thomas and by Dvořák and Kawarabayashi, respectively. Using results on hyperbolic families due to Postle and Thomas, we show that this implies that there exists a universal constant <em>c</em> such that if Σ is a surface of Euler genus <em>g</em>, every graph of edge-width at least <span><math><mi>c</mi><mo>⋅</mo><mi>log</mi><mo></mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> embedded in Σ is 5-correspondence colourable. This is asymptotically best possible, and improves upon the best known edge-width bound due to Kim, Kostochka, Li, and Zhu. Using results of Dvořák and Kawarabayashi, we show further that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 29-68"},"PeriodicalIF":1.2,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.jctb.2025.06.001
József Balogh , Felix Christian Clemen , Haoran Luo
For every integer , denote by the hypergraph on vertex set with hyperedges . We determine for every and sufficiently large n and characterize the extremal -free hypergraphs. In particular, if n satisfies certain divisibility conditions, then the extremal -free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts in the partition; each part spans a -design. This generalizes earlier work of Frankl and Füredi on the Turán number of .
Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs , for , are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.
{"title":"Non-degenerate hypergraphs with exponentially many extremal constructions","authors":"József Balogh , Felix Christian Clemen , Haoran Luo","doi":"10.1016/j.jctb.2025.06.001","DOIUrl":"10.1016/j.jctb.2025.06.001","url":null,"abstract":"<div><div>For every integer <span><math><mi>t</mi><mo>⩾</mo><mn>0</mn></math></span>, denote by <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> the hypergraph on vertex set <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>5</mn><mo>+</mo><mi>t</mi><mo>}</mo></math></span> with hyperedges <span><math><mo>{</mo><mn>123</mn><mo>,</mo><mn>124</mn><mo>}</mo><mo>∪</mo><mo>{</mo><mn>34</mn><mi>k</mi><mo>:</mo><mn>5</mn><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mn>5</mn><mo>+</mo><mi>t</mi><mo>}</mo></math></span>. We determine <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>)</mo></math></span> for every <span><math><mi>t</mi><mo>⩾</mo><mn>0</mn></math></span> and sufficiently large <em>n</em> and characterize the extremal <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span>-free hypergraphs. In particular, if <em>n</em> satisfies certain divisibility conditions, then the extremal <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span>-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts <span><math><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> in the partition; each part <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> spans a <span><math><mo>(</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-design. This generalizes earlier work of Frankl and Füredi on the Turán number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>:</mo><mo>=</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>.</div><div>Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow><mrow><mn>6</mn><mi>t</mi></mrow></msubsup></math></span>, for <span><math><mi>t</mi><mo>⩾</mo><mn>1</mn></math></span>, are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 1-28"},"PeriodicalIF":1.2,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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