Combinatorica最新文献

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.

The defective chromatic number of a graph class is the infimum k such that there exists an integer d such that every graph in this class can be partitioned into at most k induced subgraphs with maximum degree at most d. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger’s conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture.

In this paper we prove that every totally real algebraic integer (lambda ) of degree (d ge 2) occurs as an eigenvalue of some tree of height at most (d(d+1)/2+3). In order to prove this, for a given algebraic number (alpha ne 0), we investigate an additive semigroup that contains zero and is closed under the map (x mapsto alpha /(1-x)) for (x ne 1). The problem of finding the smallest such semigroup seems to be of independent interest.

Abstract

Given a (thick) irreducible spherical building (Omega ) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of (Omega ) (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type ({{textsf{A}}}_n) , and the case of type (mathsf {F_{4,4}}) over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).

For strongly connected, pure n-dimensional regular CW-complexes, we show that evenness (each ((n{-}1))-cell is contained in an even number of n-cells) is equivalent to generalizations of both cycle decomposition and traversability.

Abstract

We show that every graph with pathwidth strictly less than a that contains no path on (2^b) vertices as a subgraph has treedepth at most 10ab. The bound is best possible up to a constant factor.

For (0 le t le r) let m(t, r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t, r) known, for instance (m(3,3)=m(4,r)=1), and (m(t,r) le r-2) for all (t not = 5), and (m(t,r) le r-3) if r is even. We prove that (m(2l,r) le 3l - 6) for every (l ge 3) and (r ge 2 l).

A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two ({mathbb {Z}}/2)- spaces is equal to the minimum of their ({mathbb {Z}}/2)-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for G-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of G-spaces. More precisely, we show that this conjecture can possibly survive if the group G is either a cyclic p-group or a generalized quaternion group whose size is a power of 2.