Let H denote a finite simple hypergraph. The cover ideal of H, denoted by J = J(H), is the monomial ideal whose minimal generators correspond to the minimal vertex covers of H. We give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J . Furthermore, we study the sets Ass(R/Js) by exploring th...

In a graph G = (V; E), a set of vertices S is nearly perfect if every vertex in V ? S is adjacent to at most one vertex in S. Nearly perfect sets are closely related to 2-packings of graphs, strongly stable sets, dominating sets and eecient dominating sets. We say a nearly perfect set S is 1-minimal if for every vertex u in S, the set S ? fug is not nearly perfect. Similarly, a nearly perfect s...

A new way of constructing (minimal) perfect hash functions is described. The technique considerably reduces the overhead associated with resolving buckets in two-level hashing schemes. Evaluating a hash function requires just one multiplication and a few additions apart from primitive bit operations. The number of accesses to memory is two, one of which is to a fixed location. This improves the...

We consider the problem of covering the complete r-uniform hypergraphs on n vertices using complete r-partite graphs. We obtain lower bounds on the size of such a covering. For small values of r our result implies a lower bound of Ω( e r r √ r n log n) on the size of any such covering. This improves the previous bound of Ω(rn log n) due to Snir [5]. We also obtain good lower bounds on the size ...

We prove that every n-vertex cubic bridgeless graph has at least n/2 perfect matchings and give a list of all 17 such graphs that have less than n/2 + 2 perfect matchings.

An odd perfect number, N , is shown to have at least nine distinct prime factors. If 3 N then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.

Georges Voronoi (1908-09) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with L-type domains. A form is perfect if it can be reconstructed from all representations of its arithmetic minimum. Two forms have the same L-type if Delaunay tilings of their lattices are affinely equivalent. Delaunay (1937-38) asked about pos...

In the random k-uniform hypergraph Hk(n, p) on a vertex set V of size n, each subset of size k of V independently belongs to it with probability p. Motivated by a theorem of Erdős and Rényi [6] regarding when a random graph G(n, p) = H2(n, p) has a perfect matching, Schmidt and Shamir [14] essentially conjectured the following. Conjecture Let k|n for fixed k ≥ 3, and the expected degree d(n, p)...

We study the Lovász-Schrijver SDP-operator applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the SDP-operator generates the stable set polytope in one step has been open since 1990. In an earlier publication, we named these graphs N+-perfect. In the current contribution, we propose a conjecture on combinatorial...

Binary 1-perfect codes which give rise to partitions of the n-cube are presented. The 1-perfect partitions are characterized as homomorphic images of simple algebraic structures on F and are constructed starting from a particular case of a structure defined in F. A special property (so-called well-ordering) of STS(n) is given in such a way that for this kind of STS it is possible to define the ...