Combinatorica最新文献

{"title":"Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods","authors":"Felix Klingelhoefer, Alantha Newman","doi":"10.1007/s00493-024-00098-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00098-z","url":null,"abstract":"<p>The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc <i>uv</i> in a tournament <i>T</i> is the set of vertices that form a directed triangle with arc <i>uv</i>. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erdős and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solution to a Problem of Grünbaum on the Edge Density of 4-Critical Planar Graphs","authors":"Zdeněk Dvořák, Carl Feghali","doi":"10.1007/s00493-024-00100-8","DOIUrl":"https://doi.org/10.1007/s00493-024-00100-8","url":null,"abstract":"<p>We show that <span>(limsup |E(G)|/|V(G)| = 2.5)</span> over all 4-critical planar graphs <i>G</i>, answering a question of Grünbaum from 1988.\u0000</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"74 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Turán Density of Long Tight Cycle Minus One Hyperedge","authors":"József Balogh, Haoran Luo","doi":"10.1007/s00493-024-00099-y","DOIUrl":"https://doi.org/10.1007/s00493-024-00099-y","url":null,"abstract":"<p>Denote by <span>({mathcal {C}}^-_{ell })</span> the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on <span>(ell )</span> vertices. It is conjectured that the Turán density of <span>({mathcal {C}}^-_{5})</span> is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of <span>({mathcal {C}}^-_{ell })</span> is 1/4, for every sufficiently large <span>(ell )</span> not divisible by 3. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"301 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Small Subgraphs with Large Average Degree","authors":"Oliver Janzer, Benny Sudakov, István Tomon","doi":"10.1007/s00493-024-00091-6","DOIUrl":"https://doi.org/10.1007/s00493-024-00091-6","url":null,"abstract":"<p>In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number <span>(s&gt;2)</span>, we prove that every graph on <i>n</i> vertices with average degree <span>(dge s)</span> contains a subgraph of average degree at least <i>s</i> on at most <span>(nd^{-frac{s}{s-2}}(log d)^{O_s(1)})</span> vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with <i>n</i> vertices and average degree at least <span>(n^{1-frac{2}{s}+varepsilon })</span> contains a subgraph of average degree at least <i>s</i> on <span>(O_{varepsilon ,s}(1))</span> vertices, which is also optimal up to the constant hidden in the <i>O</i>(.) notation, and resolves a conjecture of Verstraëte.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"55 9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs","authors":"Alexandr Kostochka, Ruth Luo, Grace McCourt","doi":"10.1007/s00493-024-00096-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00096-1","url":null,"abstract":"<p>Dirac proved that each <i>n</i>-vertex 2-connected graph with minimum degree at least <i>k</i> contains a cycle of length at least <span>(min {2k, n})</span>. We consider a hypergraph version of this result. A <i>Berge cycle</i> in a hypergraph is an alternating sequence of distinct vertices and edges <span>(v_1,e_2,v_2, ldots , e_c, v_1)</span> such that <span>({v_i,v_{i+1}} subseteq e_i)</span> for all <i>i</i> (with indices taken modulo <i>c</i>). We prove that for <span>(n ge k ge r+2 ge 5)</span>, every 2-connected <i>r</i>-uniform <i>n</i>-vertex hypergraph with minimum degree at least <span>({k-1 atopwithdelims ()r-1} + 1)</span> has a Berge cycle of length at least <span>(min {2k, n})</span>. The bound is exact for all <span>(kge r+2ge 5)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Globally Linked Pairs of Vertices in Generic Frameworks","authors":"Tibor Jordán, Soma Villányi","doi":"10.1007/s00493-024-00094-3","DOIUrl":"https://doi.org/10.1007/s00493-024-00094-3","url":null,"abstract":"<p>A <i>d</i>-dimensional framework is a pair (<i>G</i>, <i>p</i>), where <span>(G=(V,E))</span> is a graph and <i>p</i> is a map from <i>V</i> to <span>({mathbb {R}}^d)</span>. The length of an edge <span>(xyin E)</span> in (<i>G</i>, <i>p</i>) is the distance between <i>p</i>(<i>x</i>) and <i>p</i>(<i>y</i>). A vertex pair <span>({u,v})</span> of <i>G</i> is said to be globally linked in (<i>G</i>, <i>p</i>) if the distance between <i>p</i>(<i>u</i>) and <i>p</i>(<i>v</i>) is equal to the distance between <i>q</i>(<i>u</i>) and <i>q</i>(<i>v</i>) for every <i>d</i>-dimensional framework (<i>G</i>, <i>q</i>) in which the corresponding edge lengths are the same as in (<i>G</i>, <i>p</i>). We call (<i>G</i>, <i>p</i>) globally rigid in <span>({mathbb {R}}^d)</span> when each vertex pair of <i>G</i> is globally linked in (<i>G</i>, <i>p</i>). A pair <span>({u,v})</span> of vertices of <i>G</i> is said to be weakly globally linked in <i>G</i> in <span>({mathbb {R}}^d)</span> if there exists a generic framework (<i>G</i>, <i>p</i>) in which <span>({u,v})</span> is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a <span>((d+1))</span>-connected graph <i>G</i> in <span>({mathbb {R}}^d)</span> and then show that for <span>(d=2)</span> it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in <span>({mathbb {R}}^2)</span>, which gives rise to an algorithm for testing weak global linkedness in the plane in <span>(O(|V|^2))</span> time. Our methods lead to a new short proof for the characterization of globally rigid graphs in <span>({mathbb {R}}^2)</span>, and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"26 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounded-Diameter Tree-Decompositions","authors":"Eli Berger, Paul Seymour","doi":"10.1007/s00493-024-00088-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00088-1","url":null,"abstract":"<p>When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to tie these two areas together. We will prove that there is a tree-decomposition in which each bag has small diameter, if and only if there is a map <span>(phi )</span> from <i>V</i>(<i>G</i>) into the vertex set of a tree <i>T</i>, such that for all <span>(u,vin V(G))</span>, the distances <span>(d_G(u,v), d_T(phi (u),phi (v)))</span> differ by at most a constant. A necessary condition for admitting such a tree-decomposition is that there is no long geodesic cycle, and for graphs of bounded tree-width, Diestel and Müller showed that this is also sufficient. But it is not sufficient in general, even qualitatively, because there are graphs in which every geodesic cycle has length at most three, and yet every tree-decomposition has a bag with large diameter. There is a more general necessary condition, however. A “geodesic loaded cycle” in <i>G</i> is a pair (<i>C</i>, <i>F</i>), where <i>C</i> is a cycle of <i>G</i> and <span>(Fsubseteq E(C))</span>, such that for every pair <i>u</i>, <i>v</i> of vertices of <i>C</i>, one of the paths of <i>C</i> between <i>u</i>, <i>v</i> contains at most <span>(d_G(u,v))</span> <i>F</i>-edges, where <span>(d_G(u,v))</span> is the distance between <i>u</i>, <i>v</i> in <i>G</i>. We will show that a (possibly infinite) graph <i>G</i> admits a tree-decomposition in which every bag has small diameter, if and only if |<i>F</i>| is small for every geodesic loaded cycle (<i>C</i>, <i>F</i>). Our proof is an extension of an algorithm to approximate tree-length in finite graphs by Dourisboure and Gavoille. In metric geometry, there is a similar theorem that characterizes when a graph is quasi-isometric to a tree, “Manning’s bottleneck criterion”. The goal of this paper is to tie all these concepts together, and add a few more related ideas. For instance, we prove a conjecture of Rose McCarty, that <i>G</i> admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices <i>u</i>, <i>v</i>, <i>w</i> of <i>G</i>, some ball of small radius meets every path joining two of <i>u</i>, <i>v</i>, <i>w</i>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"47 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 VIII: Excluding a Forest in (Theta, Prism)-Free Graphs","authors":"Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl","doi":"10.1007/s00493-024-00097-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00097-0","url":null,"abstract":"<p>Given a graph <i>H</i>, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to <i>H</i>, if and only if <i>H</i> is a forest.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree","authors":"Shirshendu Ganguly, Ella Hiesmayr, Kyeongsik Nam","doi":"10.1007/s00493-024-00086-3","DOIUrl":"https://doi.org/10.1007/s00493-024-00086-3","url":null,"abstract":"<p>Let <i>N</i> be the number of triangles in an Erdős–Rényi graph <span>({mathcal {G}}(n,p))</span> on <i>n</i> vertices with edge density <span>(p=d/n,)</span> where <span>(d&gt;0)</span> is a fixed constant. It is well known that <i>N</i> weakly converges to the Poisson distribution with mean <span>({d^3}/{6})</span> as <span>(nrightarrow infty )</span>. We address the upper tail problem for <i>N</i>, namely, we investigate how fast <i>k</i> must grow, so that <span>({mathbb {P}}(Nge k))</span> is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when <span>(k^{1/3} log k&lt; (frac{3}{sqrt{2}} - {o(1)})^{2/3} log n)</span> (sub-critical regime) as well as pin down the tail behavior when <span>(k^{1/3} log k&gt; (frac{3}{sqrt{2}} + {o(1)})^{2/3} log n)</span> (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost <i>k</i> vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately <span>((6k)^{1/3})</span>. This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades and culminating in Harel et al. (Duke Math J 171(10):2089–2192, 2022), which analyzed the problem only in the regime <span>(pgg frac{1}{n}.)</span> The proofs rely on several novel graph theoretical results which could have other applications.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"63 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140346405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Number of Topological Types of Trees","authors":"","doi":"10.1007/s00493-024-00087-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00087-2","url":null,"abstract":"<h3>Abstract</h3> <p>Two graphs are of the same <em>topological type</em> if they can be mutually embedded into each other topologically. We show that there are exactly <span> <span>(aleph _1)</span> </span> distinct topological types of countable trees. In general, for any infinite cardinal <span> <span>(kappa )</span> </span> there are exactly <span> <span>(kappa ^+)</span> </span> distinct topological types of trees of size <span> <span>(kappa )</span> </span>. This solves a problem of van der Holst from 2005.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140346417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}