Perfect matchings in large uniform hypergraphs with large minimum collective degree

We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of ⌊n/k⌋ disjoint edges. Let δk−1(H) be the largest integer d such that every (k−1)-element set of vertices of H belongs to at least d edges of H.In this paper we study the relation between δk−1(H) and the presence of a perfect matching in H for k?3. Let t(k,n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H)?t contains a perfect matching.For large n divisible by k, we completely determine the values of t(k,n), which turn out to be very close to n/2−k. For example, if k is odd and n is large and even, then t(k,n)=n/2−k+2. In contrast, for n not divisible by k, we show that t(k,n)∼n/k.In the proofs we employ a newly developed “absorbing” technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.     

On Turan hypergraphs

Let α(H) be the stability number of a hypergraph H = (X, $\text{E}$). T(n, k, α) is the smallest q such that there exists a k-uniform hypergraph H with n vertices, q edges and with α(H) ? α. A k-uniform hypergraph H, with n vertices, T(n, k, α) edges and α(H) ?α is a Turan hypergraph. The value of T(n, 2, α) is given by a theorem of Turan. In this paper new lower bounds to T(n, k, α) are obtained and it is proved that an infinity of affine spaces are Turan hypergraphs.     

Topology of random clique complexes

In a seminal paper, Erd?s and Rényi identified a sharp threshold for connectivity of the random graph G(n,p). In particular, they showed that if p?logn/n then G(n,p) is almost always connected, and if p?logn/n then G(n,p) is almost always disconnected, as n→∞.The clique complexX(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erd?s-Rényi Theorem.We study here the higher homology groups of X(G(n,p)). For k>0 we show the following. If p=n^α, with α<−1/k or α>−1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if −1/k<α<−1/(k+1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

3-uniform hypergraphs avoiding a given odd cycle

We give upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In particular, we show that a 3-uniform hypergraph containing no cycle of length 2k+1 has less than 4k ⁴ n ^1+1/k +O(n) edges. Constructions show that these bounds are best possible (up to constant factor) for k=1,2,3, 5.     

Ordering uniform supertrees by their spectral radii

A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.     

Perfect matchings in uniform hypergraphs with large minimum degree

A perfect matching in a k-uniform hypergraph on n vertices, n divisible by k, is a set of n/k disjoint edges. In this paper we give a sufficient condition for the existence of a perfect matching in terms of a variant of the minimum degree. We prove that for every k≥3 and sufficiently large n, a perfect matching exists in every n-vertex k-uniform hypergraph in which each set of k−1 vertices is contained in n/2+Ω(logn) edges. Owing to a construction in [D. Kühn, D. Osthus, Matchings in hypergraphs of large minimum degree, J. Graph Theory 51 (1) (2006) 269–280], this is nearly optimal. For almost perfect and fractional perfect matchings we show that analogous thresholds are close to n/k rather than n/2.     

Type-B generalized triangulations and determinantal ideals

For n≥3, let Ω_n be the set of line segments between the vertices of a convex n-gon. For j≥2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k≥1, let Δ_n,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing.The complex Δ_n,k has been the central object of many papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line segments in Ω_2n which can be transformed into each other by a 180^°-rotation of the 2n-gon. Let F_n be the set Ω_2n after identification, then the complex D_n,k of type-B generalized triangulations is the simplicial complex of subsets of F_n not containing any (k+1)-crossing in the above sense. For k=1, we have that D_n,1 is the simplicial complex of type-B triangulations of the 2n-gon as defined in [R. Simion, A type-B associahedron, Adv. Appl. Math. 30 (2003) 2-25] and decomposes into a join of an (n−1)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that D_n,k is a pure, k(n−k)−1+kn dimensional complex that decomposes into a kn−1-simplex and a k(n−k)−1 dimensional homology-sphere. For k=n−2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of D_n,k.On the algebraical side we give a term order on the monomials in the variables X_ij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1) times (k+1) minors of the generic n×n matrix contains the Stanley-Reisner ideal of D_n,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k<n.     

Shelling Coxeter-like complexes and sorting on trees

In their work on ‘Coxeter-like complexes’, Babson and Reiner introduced a simplicial complex ΔT associated to each tree T on n nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that ΔT is (n−b−1)-connected when the tree has b leaves. We provide a shelling for the (n−b)-skeleton of ΔT, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree T which imply shellability of ΔT, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes Mm,n with n?2m−1. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.     

Homology representations arising from the half cube

We construct a CW decomposition Cn of the n-dimensional half cube in a manner compatible with its structure as a polytope. For each 3?k?n, the complex Cn has a subcomplex Cn,k, which coincides with the clique complex of the half cube graph if k=4. The homology of Cn,k is concentrated in degree k−1 and furthermore, the (k−1)st Betti number of Cn,k is equal to the (k−2)nd Betti number of the complement of the k-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type Dn act naturally on the complexes Cn,k, and thus on the associated homology groups.     

On Samelson products in p-localized unitary groups

The unitary group U(n) has elements εi∈π2i+1(U(n)) (0?i?n−1) of its homotopy groups in the stable range. In this paper we show that certain multi Samelson products of type 〈εi,〈εj,εk〉〉 are non-trivial. This leads us to the result that the nilpotency class of the group of the self homotopy set [SU(n),SU(n)] is no less than 3, if 4?n. Also by the power of generalized Samelson products, we can see the further result that, for a prime p and an integer n=pk, nil[SU(n),SU(n)]_(p)?3, if (1) p?7 or (2) p=5 and n≡0 or 1mod4.     

On the topology of two partition posets with forbidden block sizes

We study two subposets of the partition lattice obtained by restricting block sizes. The first consists of set partitions of {1,…,n} with block size at most k, for k≤n−2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2k+2≥n and n=3k+2. For 2k+2>n, the posets in fact have the same S_n−1-homotopy type as the order complex of Π_n−1, and the S_n-homology representation is the “tree representation” of Robinson and Whitehouse. We present similar results for the subposet of Π_n in which a unique block size k≥3 is forbidden. For 2k≥n, the order complex has the homotopy type of a wedge of (n−4)-spheres. The homology representation of S_n can be simply described in terms of the Whitehouse lifting of the homology representation of Π_n−1.     

Coloring Non-uniform Hypergraphs Without Short Cycles

The work deals with a generalization of Erd?s–Lovász problem concerning colorings of non-uniform hypergraphs. Let H  = (V, E) be a hypergraph and let \({{f_r(H)=\sum\limits_{e \in E}r^{1-|e|}}}\) for some r ≥ 2. Erd?s and Lovász proposed to find the value f (n) equal to the minimum possible value of f ₂(H) where H is 3-chromatic hypergraph with minimum edge-cardinality n. In the paper we study similar problem for the class of hypergraphs with large girth. We prove that if H is a hypergraph with minimum edge-cardinality n ≥ 3 and girth at least 4, satisfying the inequality $$f_r(H) \leq \frac{1}{2}\, \left(\frac{n}{{\rm ln}\, n}\right)^{2/3},$$ then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.     

On the homotopy equivalence of the spaces of proper and local maps

We prove that for n > 1 the space of proper maps P ₀(n, k) and the space of local maps F ₀(n, k) are not homotopy equivalent.     

On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G,n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n−1. In case G is an n-dimensional group there is a unique (up to homotopy) (G,n)-complex on the minimal Euler-characteristic level χmin(G,n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G,n)-complexes on the level χmin(G,n)+1. In the case where n=2 these examples are algebraic.     

Supersaturated graphs and hypergraphs

We shall consider graphs (hypergraphs) without loops and multiple edges. Let ? be a family of so called prohibited graphs and ex (n, ?) denote the maximum number of edges (hyperedges) a graph (hypergraph) onn vertices can have without containing subgraphs from ?. A graph (hyper-graph) will be called supersaturated if it has more edges than ex (n, ?). IfG hasn vertices and ex (n, ?)+k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies ofL ε ? must occur in a graphG ⁿ onn vertices with ex (n, ?)+k edges (hyperedges)?     

Higher-order recurrences for Bernoulli numbers

Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as n(B₀+B₀)=−nBn−1−(n−1)Bn, is extended to n(Bk₁+?+Bkm) for m?2 and arbitrary fixed integers k₁,…,km?0. In the general case we prove an existence theorem for Euler-type formulas, and for m=3 we obtain explicit expressions. This extends the authors' previous work for m=2.