Pub Date : 2025-09-11DOI: 10.1016/j.jctb.2025.08.006
Paul Seymour
Let G be an acyclic digraph, and let , where are sources, are sinks, and every other vertex has in-degree and out-degree at least two. In 1985, Thomassen showed that there do not exist disjoint directed paths from a to c and from b to d, if and only if G can be drawn in a closed disc with drawn in the boundary in order. We give a shorter proof.
{"title":"Thomassen's theorem on the two-linkage problem in acyclic digraphs: A shorter proof","authors":"Paul Seymour","doi":"10.1016/j.jctb.2025.08.006","DOIUrl":"10.1016/j.jctb.2025.08.006","url":null,"abstract":"<div><div>Let <em>G</em> be an acyclic digraph, and let <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> are sources, <span><math><mi>c</mi><mo>,</mo><mi>d</mi></math></span> are sinks, and every other vertex has in-degree and out-degree at least two. In 1985, Thomassen showed that there do not exist disjoint directed paths from <em>a</em> to <em>c</em> and from <em>b</em> to <em>d</em>, if and only if <em>G</em> can be drawn in a closed disc with <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></math></span> drawn in the boundary in order. We give a shorter proof.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 97-100"},"PeriodicalIF":1.2,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049038","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 : 2025-09-08DOI: 10.1016/j.jctb.2025.08.003
Maria Chudnovsky , Sepehr Hajebi , Daniel Lokshtanov , Sophie Spirkl
A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree independence number. More explicitly, we show that there exists a constant c such that every n-vertex 3PC-free graph has a tree decomposition in which every bag has stability number at most . This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.
{"title":"Tree independence number II. Three-path-configurations","authors":"Maria Chudnovsky , Sepehr Hajebi , Daniel Lokshtanov , Sophie Spirkl","doi":"10.1016/j.jctb.2025.08.003","DOIUrl":"10.1016/j.jctb.2025.08.003","url":null,"abstract":"<div><div>A <em>three-path-configuration</em> is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is <em>3PC-free</em> if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree independence number. More explicitly, we show that there exists a constant <em>c</em> such that every <em>n</em>-vertex 3PC-free graph has a tree decomposition in which every bag has stability number at most <span><math><mi>c</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>. This implies that the <span>Maximum Weight Independent Set</span> problem, as well as several other natural algorithmic problems, that are known to be <span>NP</span>-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 74-96"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010349","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 : 2025-08-28DOI: 10.1016/j.jctb.2025.08.002
Maria Chudnovsky , Linda Cook , James Davies , Sang-il Oum
A class of graphs is χ-bounded if there is a function f such that for all induced subgraphs H of a graph in . If f can be chosen to be a polynomial, we say that is polynomially χ-bounded. Esperet proposed a conjecture that every χ-bounded class of graphs is polynomially χ-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are χ-bounded but not polynomially χ-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class of graphs is Pollyanna if is polynomially χ-bounded for every χ-bounded class of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.
{"title":"Reuniting χ-boundedness with polynomial χ-boundedness","authors":"Maria Chudnovsky , Linda Cook , James Davies , Sang-il Oum","doi":"10.1016/j.jctb.2025.08.002","DOIUrl":"10.1016/j.jctb.2025.08.002","url":null,"abstract":"<div><div>A class <span><math><mi>F</mi></math></span> of graphs is <em>χ</em>-bounded if there is a function <em>f</em> such that <span><math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>)</mo></math></span> for all induced subgraphs <em>H</em> of a graph in <span><math><mi>F</mi></math></span>. If <em>f</em> can be chosen to be a polynomial, we say that <span><math><mi>F</mi></math></span> is polynomially <em>χ</em>-bounded. Esperet proposed a conjecture that every <em>χ</em>-bounded class of graphs is polynomially <em>χ</em>-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are <em>χ</em>-bounded but not polynomially <em>χ</em>-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class <span><math><mi>C</mi></math></span> of graphs is Pollyanna if <span><math><mi>C</mi><mo>∩</mo><mi>F</mi></math></span> is polynomially <em>χ</em>-bounded for every <em>χ</em>-bounded class <span><math><mi>F</mi></math></span> of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 30-73"},"PeriodicalIF":1.2,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907692","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 : 2025-08-20DOI: 10.1016/j.jctb.2025.07.008
Ervin Győri , Binlong Li , Nika Salia , Casey Tompkins , Kitti Varga , Manran Zhu
Bollobás proved that for every k and ℓ such that contains an even number, an n-vertex graph containing no cycle of length can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs ℓ and k. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length .
Bollobás证明了对于每一个k和r，使得k z + r包含一个偶数，一个n顶点的图，不包含长度为r modk的循环，最多只能包含一个线性数的边。这种图中最大边数的精确（或渐近）值对于很少的对（r和k）是已知的。在这项工作中，我们精确地确定了不包含长度为0mod4的循环的图中的最大边数。
{"title":"On graphs without cycles of length 0 modulo 4","authors":"Ervin Győri , Binlong Li , Nika Salia , Casey Tompkins , Kitti Varga , Manran Zhu","doi":"10.1016/j.jctb.2025.07.008","DOIUrl":"10.1016/j.jctb.2025.07.008","url":null,"abstract":"<div><div>Bollobás proved that for every <em>k</em> and <em>ℓ</em> such that <span><math><mi>k</mi><mi>Z</mi><mo>+</mo><mi>ℓ</mi></math></span> contains an even number, an <em>n</em>-vertex graph containing no cycle of length <span><math><mi>ℓ</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>k</mi></math></span> can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs <em>ℓ</em> and <em>k</em>. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length <span><math><mn>0</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 7-29"},"PeriodicalIF":1.2,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863500","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 : 2025-08-08DOI: 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}
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