Related papers: The existence of square non-integer Heffter arrays

An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering…

A hypergraph is a $T_0$-hypergraph if for every two different vertices of the hypergraph there exists an edge containing one of the vertices and not containing the other. A general method for the enumeration of certain classes of…

Let $f(n)$ denote the maximum sum of the side lengths of $n$ non-overlapping squares packed inside a unit square. We prove that $f(n^2+1) = n$ for all positive integers $n$ if and only if the sum $\sum_{k\geq 1}(f(k^2+1)-k)$ converges. We…

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of…

Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\ldots,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\cup B_i$ and…

We consider the pattern detection problem in graphs: given a constant size pattern graph $H$ and a host graph $G$, determine whether $G$ contains a subgraph isomorphic to $H$. Our main results are: * We prove that if a pattern $H$ contains…

Let $\UT_n(\FF_q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of $\UT_{n+1}(\FF_q)$ are indexed by lattice paths from the origin to the…

We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a…

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$…

An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…

Finite frame theory has become a powerful tool for many applications of mathematics. In this paper we introduce a new area of research in frame theory: Integer frames. These are frames having all integer coordinates with respect to a fixed…

An integer packing set is a set of non-negative integer vectors with the property that, if a vector $x$ is in the set, then every non-negative integer vector $y$ with $y \leq x$ is in the set as well. Integer packing sets appear naturally…

A graph is Hamiltonian if it contains a cycle which visits every vertex of the graph exactly once. In this paper, we consider the problem of Hamiltonicity of a graph $G_n$, which will be called the prime difference graph of order $n$, with…

A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove…

Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc…

Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this…

Let $M$ be a compact hyperkaehler manifold. The hyperkaehler structure equips $M$ with a set $R$ of complex structures parametrized by $CP^1$, called "the set of induced complex structures". It was known previously that induced complex…

For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a perfect matching for any $Q\subseteq V(G)$ with $|Q|=k$. In this article, we prove spectral radius conditions for the existence of…

Let $H$ be a random $k$-uniform $n$-vertex hypergraph where every $k$-tuple belongs to $H$ independently with probability $p$. We show that for some $\varepsilon_k > 0$, if $p \geq n^{-\varepsilon_k}$, then asymptotically almost surely $H$…

A {\em signed magic rectangle} $SMR(m,n;k, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (mk-1)/2\}$ if $mk$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mk/2\}$ if $mk$ is even, such that precisely $k$…