Lucas Aragão , João Pedro Marciano , Walner Mendonça
{"title":"Degree conditions for Ramsey goodness of paths","authors":"Lucas Aragão , João Pedro Marciano , Walner Mendonça","doi":"10.1016/j.ejc.2024.104082","DOIUrl":"10.1016/j.ejc.2024.104082","url":null,"abstract":"<div><div>A classical result of Chvátal implies that if <span><math><mrow><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>, then any colouring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in red and blue contains either a monochromatic red <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> or a monochromatic blue <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>. We study a natural generalisation of his result, determining the exact minimum degree condition for a graph <span><math><mi>G</mi></math></span> on <span><math><mrow><mi>n</mi><mo>=</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> vertices which guarantees that the same Ramsey property holds in <span><math><mi>G</mi></math></span>. In particular, using a slight generalisation of a result of Haxell, we show that <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mfenced><mrow><mi>t</mi><mo>/</mo><mn>2</mn></mrow></mfenced></mrow></math></span> suffices, and that this bound is best possible. We also use a classical result of Bollobás, Erdős, and Straus to prove a tight minimum degree condition in the case <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>t</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104082"},"PeriodicalIF":1.0,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the faces of unigraphic 3-polytopes","authors":"Riccardo W. Maffucci","doi":"10.1016/j.ejc.2024.104081","DOIUrl":"10.1016/j.ejc.2024.104081","url":null,"abstract":"<div><div>A 3-polytope is a 3-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other 3-polytope, up to graph isomorphism. The classification of unigraphic 3-polytopes appears to be a difficult problem.</div><div>In this paper we prove that, apart from pyramids, all unigraphic 3-polytopes have no <span><math><mi>n</mi></math></span>-gonal faces for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>10</mn></mrow></math></span>. Our method involves defining several planar graph transformations on a given 3-polytope containing an <span><math><mi>n</mi></math></span>-gonal face with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>10</mn></mrow></math></span>. The delicate part is to prove that, for every such 3-polytope, at least one of these transformations both preserves 3-connectivity, and is not an isomorphism.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104081"},"PeriodicalIF":1.0,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441010","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":"Bounded unique representation bases for the integers","authors":"Yong-Gao Chen, Jin-Hui Fang","doi":"10.1016/j.ejc.2024.104080","DOIUrl":"10.1016/j.ejc.2024.104080","url":null,"abstract":"<div><div>For a nonempty set <span><math><mi>A</mi></math></span> of integers and an integer <span><math><mi>n</mi></math></span>, let <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of representations of <span><math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> with <span><math><mrow><mi>a</mi><mo>≤</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math></span>, and let <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of representations of <span><math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> with <span><math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math></span>. Erdős and Turán (1941) posed the profound conjecture: if <span><math><mi>A</mi></math></span> is a set of positive integers such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span> for all sufficiently large <span><math><mi>n</mi></math></span>, then <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is unbounded. Nešetřil and Serra (2004) introduced the notion of bounded sets and confirmed the Erdős–Turán conjecture for all bounded bases. Nathanson (2003) considered the existence of the set <span><math><mi>A</mi></math></span> with logarithmic growth such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all integers <span><math><mi>n</mi></math></span>. In this paper, we prove that, for any positive function <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mn>0</mn></mrow></math></span> as <span><math><mrow><mi>x</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, there is a bounded set <span><math><mi>A</mi></math></span> of integers such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all integers <span><math><mi>n</mi></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all positi","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104080"},"PeriodicalIF":1.0,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441009","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}
Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour
{"title":"Induced subgraph density. II. Sparse and dense sets in cographs","authors":"Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour","doi":"10.1016/j.ejc.2024.104075","DOIUrl":"10.1016/j.ejc.2024.104075","url":null,"abstract":"<div><div>A well-known theorem of Rödl says that for every graph <span><math><mi>H</mi></math></span>, and every <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, there exists <span><math><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math></span> such that if <span><math><mi>G</mi></math></span> does not contain an induced copy of <span><math><mi>H</mi></math></span>, then there exists <span><math><mrow><mi>X</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>δ</mi><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span> such that one of <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span> has edge-density at most <span><math><mi>ɛ</mi></math></span>. But how does <span><math><mi>δ</mi></math></span> depend on <span><math><mi>ϵ</mi></math></span>? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all <span><math><mi>H</mi></math></span> there exists <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for all <span><math><mi>ɛ</mi></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span>, Rödl’s theorem holds with <span><math><mrow><mi>δ</mi><mo>=</mo><msup><mrow><mi>ɛ</mi></mrow><mrow><mi>c</mi></mrow></msup></mrow></math></span>. This conjecture implies the Erdős–Hajnal conjecture, and until now it had not been verified for any non-trivial graphs <span><math><mi>H</mi></math></span>. Our first result shows that it is true when <span><math><mrow><mi>H</mi><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math></span>. Indeed, in that case we can take <span><math><mrow><mi>δ</mi><mo>=</mo><mi>ɛ</mi></mrow></math></span>, and insist that one of <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span> has maximum degree at most <span><math><mrow><msup><mrow><mi>ɛ</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span>).</div><div>Second, we will show that every graph <span><math><mi>H</mi></math></span> that can be obtained by substitution from copies of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> satisfies the Fox–Sudakov conjecture. To prove this, we need to work with a stronger property. Let us say <span><math><mi>H</mi></math></span> is <em>viral</em> if there exists <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for all <span><math><mi>ɛ</mi></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104075"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423576","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}
Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano
{"title":"The diameter of randomly twisted hypercubes","authors":"Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano","doi":"10.1016/j.ejc.2024.104078","DOIUrl":"10.1016/j.ejc.2024.104078","url":null,"abstract":"<div><div>The <span><math><mi>n</mi></math></span>-dimensional random twisted hypercube <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is constructed recursively by taking two instances of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>/</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with high probability and at least <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></math></span>. We improve their upper bound by showing that <span><math><mrow><mi>diam</mi><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></mfrac></mrow></math></span> with high probability.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104078"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423587","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}
Ademir Hujdurović , István Kovács , Klavdija Kutnar , Dragan Marušič
{"title":"Intersection density of transitive groups with small cyclic point stabilizers","authors":"Ademir Hujdurović , István Kovács , Klavdija Kutnar , Dragan Marušič","doi":"10.1016/j.ejc.2024.104079","DOIUrl":"10.1016/j.ejc.2024.104079","url":null,"abstract":"<div><div>For a permutation group <span><math><mi>G</mi></math></span> acting on a set <span><math><mi>V</mi></math></span>, a subset <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span> is said to be an <em>intersecting set</em> if for every pair of elements <span><math><mrow><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>F</mi></mrow></math></span> there exists <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> such that <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. The <em>intersection density</em> <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a transitive permutation group <span><math><mi>G</mi></math></span> is the maximum value of the quotient <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>/</mo><mrow><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span> where <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> is a stabilizer of a point <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></math></span> and <span><math><mi>F</mi></math></span> runs over all intersecting sets in <span><math><mi>G</mi></math></span>. If <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> is a largest intersecting set in <span><math><mi>G</mi></math></span> then <span><math><mi>G</mi></math></span> is said to have the <em>Erdős-Ko-Rado (EKR)-property</em>. This paper is devoted to the study of transitive permutation groups, with point stabilizers of prime order with a special emphasis given to orders 2 and 3, which do not have the EKR-property. Among others, constructions of an infinite family of transitive permutation groups having point stabilizer of order 3 with intersection density <span><math><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></math></span> and of infinite families of transitive permutation groups having point stabilizer of order 3 with arbitrarily large intersection density are given.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104079"},"PeriodicalIF":1.0,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423575","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":"Turán numbers of ordered tight hyperpaths","authors":"John P. Bright, Kevin G. Milans, Jackson Porter","doi":"10.1016/j.ejc.2024.104070","DOIUrl":"10.1016/j.ejc.2024.104070","url":null,"abstract":"<div><div>An <em>ordered hypergraph</em> is a hypergraph <span><math><mi>G</mi></math></span> whose vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is linearly ordered. We find the Turán numbers for the <span><math><mi>r</mi></math></span>-uniform <span><math><mi>s</mi></math></span>-vertex tight path <span><math><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup></math></span> (with vertices in the natural order) exactly when <span><math><mrow><mi>r</mi><mo>≤</mo><mi>s</mi><mo><</mo><mn>2</mn><mi>r</mi></mrow></math></span> and <span><math><mi>n</mi></math></span> is even; our results imply <span><math><mrow><mover><mrow><mi>ex</mi></mrow><mo>→</mo></mover><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi><mo>−</mo><mi>r</mi></mrow></msup></mrow></mfrac><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> when <span><math><mrow><mi>r</mi><mo>≤</mo><mi>s</mi><mo><</mo><mn>2</mn><mi>r</mi></mrow></math></span>. When <span><math><mrow><mi>s</mi><mo>≥</mo><mn>2</mn><mi>r</mi></mrow></math></span>, the asymptotics of <span><math><mrow><mover><mrow><mi>ex</mi></mrow><mo>→</mo></mover><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow></mrow></math></span> remain open. For <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span>, we give a construction of an <span><math><mi>r</mi></math></span>-uniform <span><math><mi>n</mi></math></span>-vertex hypergraph not containing <span><math><msubsup><mrow><mover><mrow><mi>P</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>s</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup></math></span> which we conjecture to be asymptotically extremal.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104070"},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423708","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}
John Haslegrave , Alex Scott , Youri Tamitegama , Jane Tan
{"title":"Boundary rigidity of 3D CAT(0) cube complexes","authors":"John Haslegrave , Alex Scott , Youri Tamitegama , Jane Tan","doi":"10.1016/j.ejc.2024.104077","DOIUrl":"10.1016/j.ejc.2024.104077","url":null,"abstract":"<div><div>The boundary rigidity problem is a classical question from Riemannian geometry: if <span><math><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></math></span> is a Riemannian manifold with smooth boundary, is the geometry of <span><math><mi>M</mi></math></span> determined up to isometry by the metric <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> induced on the boundary <span><math><mrow><mi>∂</mi><mi>M</mi></mrow></math></span>? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104077"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142359298","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}
Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke
{"title":"Decks of rooted binary trees","authors":"Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke","doi":"10.1016/j.ejc.2024.104076","DOIUrl":"10.1016/j.ejc.2024.104076","url":null,"abstract":"<div><div>We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree <span><math><mi>T</mi></math></span> refers to the set (resp. multiset) of leaf-induced binary subtrees of <span><math><mi>T</mi></math></span>. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on <span><math><mi>n</mi></math></span> leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for <span><math><mi>k</mi></math></span>-universal trees, i.e., rooted binary trees that contain all <span><math><mi>k</mi></math></span>-leaf rooted binary trees as leaf-induced subtrees.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104076"},"PeriodicalIF":1.0,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318960","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":"Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole","authors":"Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl","doi":"10.1016/j.ejc.2024.104074","DOIUrl":"10.1016/j.ejc.2024.104074","url":null,"abstract":"<div><div>A <em>clock</em> is a graph consisting of an induced cycle <span><math><mi>C</mi></math></span> and a vertex not in <span><math><mi>C</mi></math></span> with at least two non-adjacent neighbours in <span><math><mi>C</mi></math></span>. We show that every clock-free graph of large treewidth contains a “basic obstruction” of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104074"},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001598/pdfft?md5=fa98c8a13265d848775c4b52beb995da&pid=1-s2.0-S0195669824001598-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315136","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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