Combinatorica最新文献

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 (phi ) from V(G) into the vertex set of a tree T, such that for all (u,vin V(G)), the distances (d_G(u,v), d_T(phi (u),phi (v))) 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 G is a pair (C, F), where C is a cycle of G and (Fsubseteq E(C)), such that for every pair u, v of vertices of C, one of the paths of C between u, v contains at most (d_G(u,v)) F-edges, where (d_G(u,v)) is the distance between u, v in G. We will show that a (possibly infinite) graph G admits a tree-decomposition in which every bag has small diameter, if and only if |F| is small for every geodesic loaded cycle (C, F). 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 G admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices u, v, w of G, some ball of small radius meets every path joining two of u, v, w.

Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if H is a forest.

Let N be the number of triangles in an Erdős–Rényi graph ({mathcal {G}}(n,p)) on n vertices with edge density (p=d/n,) where (d>0) is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean ({d^3}/{6}) as (nrightarrow infty ). We address the upper tail problem for N, namely, we investigate how fast k must grow, so that ({mathbb {P}}(Nge k)) 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 (k^{1/3} log k< (frac{3}{sqrt{2}} - {o(1)})^{2/3} log n) (sub-critical regime) as well as pin down the tail behavior when (k^{1/3} log k> (frac{3}{sqrt{2}} + {o(1)})^{2/3} log n) (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately ((6k)^{1/3}). 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 (pgg frac{1}{n}.) The proofs rely on several novel graph theoretical results which could have other applications.

Abstract

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 (L_1) in supercritical percolation on these product graphs, that is, when (p=frac{1+epsilon }{d}) , where d is the degree of the product graph and (epsilon >0) is a small enough constant. We show that typically (L_1) has edge-expansion (Omega left( frac{1}{dln d}right) ) . Furthermore, we show that (L_1) likely contains a linear-sized subgraph with vertex-expansion (Omega left( frac{1}{dln d}right) ) . These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of (L_1) as well as the typical mixing time of a lazy random walk on (L_1) . Furthermore, we show the likely existence of a cycle of length (Omega left( frac{n}{dln d}right) ) . 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 (L_1) were previously only known to be polynomial in d.

Abstract

Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly (aleph _1) distinct topological types of countable trees. In general, for any infinite cardinal (kappa ) there are exactly (kappa ^+) distinct topological types of trees of size (kappa ) . This solves a problem of van der Holst from 2005.

Colour the edges of the complete graph with vertex set ({{1, 2, dotsc , n}}) with an arbitrary number of colours. What is the smallest integer f(l, k) such that if (n > f(l,k)) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that (f(l, k) = l^{k - 1}) and proved this for (l ge (3 k)^{2 k}). We prove the conjecture for (l ge k^3 (log k)^{1 + o(1)}) and establish the general upper bound (f(l, k) le k (log k)^{1 + o(1)} cdot l^{k - 1}). This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.

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.

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only (|X(k-1)|=O(n)) points in contrast to (left( {begin{array}{c}n kend{array}}right) ) points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length n is always linear in n. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length n is always polynomial in n. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.

Let L be a finite lattice. Inspired by Ungar’s solution to the famous slopes problem, we define an Ungar move to be an operation that sends an element (xin L) to the meet of ({x}cup T), where T is a subset of the set of elements covered by x. We introduce the following Ungar game. Starting at the top element of L, two players—Atniss and Eeta—take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say L is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on L. We first prove that the number of principal order ideals in the weak order on (S_n) that are Eeta wins is (O(0.95586^nn!)). We then consider a broad class of intervals in Young’s lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type-A root posets. We also characterize and enumerate principal order ideals in Tamari lattices that are Eeta wins. Finally, we conclude with some open problems and a short discussion of the computational complexity of Ungar games.