Matrices with isomorphic diagonal hypergraphs

Let A be an n×n matrix of 0's and 1's with total support. The diagonal hypergraph of A is the hypergraph whose vertices are the positive positions of A and whose edges are the positive diagonals of A. We investigate which n×n matrices with total support have diagonal hypergraphs isomorphic to that of A, and the structure of such isomorphisms.     

Sur le nombre transversal des hypergraphes uniformes

The transversal number of a given hypergraph is the cardinality of the smallest set of vertices meeting all the edges. What is the maximal possible value of the transversal number of a r-uniform hypergraph on n vertices with maximal degree p? The problem is solved here for p = 2, by using Berge's theorem on matchings.     

Some hypergraphs and packing problems associated with matrices of 0's and 1's

We define various classes of hypergraphs associated with m × n matrices of 0's and 1's and determine for which classes the maximum cardinality of a set of pairwise disjoint edges equals the minimum cardinality of a set of nodes that cover all edges independently of the matrix.     

Matrices of zeros and ones with fixed row and column sum vectors

Let m and n be positive integers, and let R=(r₁,…,r_m) and S=(s₁,…,s_n) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matrices of 0's and 1's having r_i1's in row i ands_i 1's in column j. A number of new results are proved. The results can be also be formulated in terms of a set of bipartite graps with bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S.     

The maximum size of hypergraphs without generalized 4-cycles

Let fr(n) be the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain four distinct edges A, B, C, D with A∪B=C∪D and A∩B=C∩D=∅. This problem was stated by Erd?s [P. Erd?s, Problems and results in combinatorial analysis, Congr. Numer. 19 (1977) 3-12]. It can be viewed as a generalization of the Turán problem for the 4-cycle to hypergraphs.Let . Füredi [Z. Füredi, Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (1984) 161-168] observed that ?r?1 and conjectured that this is equality for every r?3. The best known upper bound ?r?3 was proved by Mubayi and Verstraëte [D. Mubayi, J. Verstraëte, A hypergraph extension of the bipartite Turán problem, J. Combin. Theory Ser. A 106 (2004) 237-253]. Here we improve this bound. Namely, we show that for every r?3, and ?₃?13/9. In particular, it follows that ?r→1 as r→∞.     

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.     

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

For two square matrices A, B of possibly different sizes with nonnegative integer entries, write A ≈₁ B if A = RS and B = SR for some two nonnegative integer matrices R,S. The transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics, where it has been shown by R.F. Williams to characterize the isomorphism of two topological Markov chains with transition matrices A and B. One invariant is the characteristic polynomial up to factors of λ. However, no procedure for deciding strong shift equivalence is known, even for 2×2 matrices A, B. In fact, for n × n matrices with n > 2, no nontrivial sufficient condition is known. This paper presents such a sufficient condition: that A and B are in the same component of a directed graph whose vertices are all n × n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities. For n > 2 this result gives confirmation of strong shift equivalences that previously could not be verified; for n = 2, previous results are strengthened and the structure of the directed graph is determined.     

Partitioning 3-uniform hypergraphs

Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges can be partitioned into r sets so that each set meets at least rm/(2r−1) edges. For r=3, Bollobás, Reed and Thomason proved the lower bound (1−1/e)m/3≈0.21m, which was improved to (5/9)m by Bollobás and Scott and to 0.6m by Haslegrave. In this paper, we show that any 3-uniform hypergraph with m edges can be partitioned into 3 sets, each of which meets at least 0.65m−o(m) edges.     

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.     

Pelikán's conjecture and cyclotomic cosets

The following conjecture was recently made by J. Pelikán. Let a₀ ,…, a_n be an (n + 1)-tuple of 0's and 1's; let A_k = ?_i=0^n?ka_ia_i+k for k = 0,…, n. Then if n ? 4 some A_k is even.This paper shows that Pelikán's conjecture is false for infinitely many values of n. On the other hand it is also shown that the conjecture is true for most values of n, and a characterization is given of those values of n for which it fails.     

Eigenvalue inequalities for products of matrix exponentials

Motivated by models from stochastic population biology and statistical mechanics, we proved new inequalities of the form $\text{(1) ?(e}{}^{\mathrm{A}}\text{e}{}^{\mathrm{B}}\text{)??(e}{}^{\mathrm{A+B}}\text{)}$, where A and B are n × n complex matrices, 1<n<∞, and ? is a real-valued continuous function of the eigenvalues of its matrix argument. For example, if A is essentially nonnegative, B is diagonal real, and ? is the spectral radius, then (1) holds; if in addition A is irreducible and B has at least two different diagonal elements, then the inequality (1) is strict. The proof uses Kingman's theorem on the log-convexity of the spectral radius, Lie's product formula, and perturbation theory. We conclude with conjectures.     

A matrix of permanents

If A is a matrix of order n×(n?2), n?3, denote by ā the n×n matrix whose (i,j)th entry is zero if i=j, and if i≠j, is the permanent of the submatrix of A obtained by deleting its ith and jth rows. It is shown that if A is a nonnegative n×(n?2) matrix, then ā is nonsingular if and only if A has no zero submatrix of n?1 lines. This is used to give precise consequences of the occurrence of equality in Alexandroff's inequality.     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

The Ramsey number for hypergraph cycles I

Let C_n denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v₁,…,v_n and edges v₁v₂v₃, v₃v₄v₅, v₅v₆v₇,…,v_n-1v_nv₁. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C_n, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.     

Extremals of functions on graphs with applications to graphs and hypergraphs

The method used in an article by T. S. Matzkin and E. G. Straus [Canad. J. Math.17 (1965), 533–540] is generalized by attaching nonnegative weights to t-tuples of vertices in a hypergraph subject to a suitable normalization condition. The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges. The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P. Turán's Theorem [Colloq. Math.3 (1954), 19–30: (Hungarian) Mat. Fiz. Lapok48 (1941), 436–452].     

4-Books of three pages

Let H be a 4-uniform hypergraph on an n-element vertex set V containing no 4-book of 3 pages, i.e., a hypergraph of 4 quadruples with vertices {1,2,…,7} and edges {1234,1235,1236,4567}. Then for n>n₀

     

Dominant dimensions,derived equivalences and tilting modules

We show that for every ? > 0 there exist δ > 0 and n₀ ∈ ? such that every 3-uniform hypergraph on n ≥ n₀ vertices with the property that every k-vertex subset, where k ≥ δn, induces at least \(\left( {\frac{1}{2} + \varepsilon } \right)\left( {\begin{array}{*{20}c} k \\ 3 \\ \end{array} } \right)\) edges, contains K₄^? as a subgraph, where K₄^? is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erd?s and Sós. The constant 1/4 is the best possible.     

Fast canonization of circular strings

A circular string A = (a₁,…,a_n) is a string that has n equivalent linear representations A_i = a_i,…,a_n,a₁,…,a_i?1 for i = 1,…,n. The a_i's are assumed to be well ordered. We say that A_i < A_j if the word a_i…a_na₁…a_i?1 precedes the word a_j…a_na₁…a_j?1 in the lexicographic order, A_i ? A_j if either A_i < A_j or A_i = A_j. A_i0 is a minimal representation of A if A_i0 ? A_i for all 1 ≤ i ≤ n. The index i₀ is called a minimal starting point (m.s.p.). In this paper we discuss the problem of finding m.s.p. of a given circular string. Our algorithm finds, in fact, all the m.s.p.'s of a given circular string A of length n by using at most $\text{n + ?}\text{d}\text{2}\text{?}$ comparisons. The number d denotes the difference between two successive m.s.p.'s (see Lemma 1.1) and is equal to n if A has a unique m.s.p. Our algorithm improves the result of 3n comparisons given by K. S. Booth. Only constant auxiliary storage is used.     

The 1-width of (0,1)-matrices having constant row sum 3

In this note we establish upper bounds for the 1-width of an m × n matrix of 0's and 1's having three 1's in every row and having a constant number, c, of 1's in every column. When c = 3, this upper bound is $\text{n}\text{2}$ and when c = 4 this estimate is $\text{5n}\text{9}$. In these cases the upper bound is best possible, in the sense that for every possible size there exist matrices with this maximal 1-width. The technique of proof is also used to improve the known bound for the 1-width of (0, 1)-matrices with constant line sum 4.     

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)?