Combinatorica最新文献

{"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}

{"title":"Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs","authors":"","doi":"10.1007/s00493-024-00089-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00089-0","url":null,"abstract":"<h3>Abstract</h3> <p>It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component <span> <span>(L_1)</span> </span> in supercritical percolation on these product graphs, that is, when <span> <span>(p=frac{1+epsilon }{d})</span> </span>, where <em>d</em> is the degree of the product graph and <span> <span>(epsilon &gt;0)</span> </span> is a small enough constant. We show that typically <span> <span>(L_1)</span> </span> has edge-expansion <span> <span>(Omega left( frac{1}{dln d}right) )</span> </span>. Furthermore, we show that <span> <span>(L_1)</span> </span> likely contains a linear-sized subgraph with vertex-expansion <span> <span>(Omega left( frac{1}{dln d}right) )</span> </span>. These results are best possible up to the logarithmic factor in <em>d</em>. Using these likely expansion properties, we determine, up to small polylogarithmic factors in <em>d</em>, the likely diameter of <span> <span>(L_1)</span> </span> as well as the typical mixing time of a lazy random walk on <span> <span>(L_1)</span> </span>. Furthermore, we show the likely existence of a cycle of length <span> <span>(Omega left( frac{n}{dln d}right) )</span> </span>. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of <span> <span>(L_1)</span> </span> were previously only known to be polynomial in <em>d</em>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"54 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140346436","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":"Flashes and Rainbows in Tournaments","authors":"António Girão, Freddie Illingworth, Lukas Michel, Michael Savery, Alex Scott","doi":"10.1007/s00493-024-00090-7","DOIUrl":"https://doi.org/10.1007/s00493-024-00090-7","url":null,"abstract":"<p>Colour the edges of the complete graph with vertex set <span>({{1, 2, dotsc , n}})</span> with an arbitrary number of colours. What is the smallest integer <i>f</i>(<i>l</i>, <i>k</i>) such that if <span>(n &gt; f(l,k))</span> then there must exist a monotone monochromatic path of length <i>l</i> or a monotone rainbow path of length <i>k</i>? Lefmann, Rödl, and Thomas conjectured in 1992 that <span>(f(l, k) = l^{k - 1})</span> and proved this for <span>(l ge (3 k)^{2 k})</span>. We prove the conjecture for <span>(l ge k^3 (log k)^{1 + o(1)})</span> and establish the general upper bound <span>(f(l, k) le k (log k)^{1 + o(1)} cdot l^{k - 1})</span>. This reduces the gap between the best lower and upper bounds from exponential to polynomial in <i>k</i>. We also generalise some of these results to the tournament setting.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140346435","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":"Lattice Path Matroids and Quotients","authors":"Carolina Benedetti-Velásquez, Kolja Knauer","doi":"10.1007/s00493-024-00085-4","DOIUrl":"https://doi.org/10.1007/s00493-024-00085-4","url":null,"abstract":"<p>We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"68 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140346412","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}