Let it(G) denote the number of independent sets of size t in a graph G. Levit and Mandrescu have conjectured that for all bipartite G the sequence (it(G))t≥0 (the independent set sequence of G) is unimodal. We provide evidence for this conjecture by showing that is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph G(n, n, p), and show that for an...

Conjecture 0.3 has a fairly interesting reformulation with some ”Ramsey flavor”. A set of pairwise disjoint edges e1, e2, . . . , et of G is called a connected matching of size t ([8]) if for every 1 ≤ i < j ≤ t there exists at least one edge of G connecting an endpoint of ei to an endpoint of ej . Alfréd Rényi Institute, Hungarian Academy of Sciences, Budapest 1364, P. O. Box 127 and Dept. of ...

In a group testing scheme, a set of tests is designed to identify a small number t of defective items among a large set (of size N ) of items. In the non-adaptive scenario the set of tests has to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a disjunct matrix, an M × N matrix where the union of supports of any t columns does not contai...

Let 1 ≤ s < t, N ≥ 1 be integers and a complex electronic circuit of size t is said to be an s-active, s ≪ t, and can work as a system block if not more than s elements of the circuit are defective. Otherwise, the circuit is said to be an s-defective and should be substituted for the s-active circuit. Suppose that there exists a possibility to check the s-activity of the circuit using N non-ada...

Let 1 ≤ s < t, N ≥ 1 be integers and a complex electronic circuit of size t is said to be an s-active, s ≪ t, and can work as a system block if not more than s elements of the circuit are defective. Otherwise, the circuit is said to be an s-defective and should be replaced by a similar s-active circuit. Suppose that there exists a possibility to run N non-adaptive group tests to check the s-act...

We study the fluctuations of self-intersection counts of random geodesic segments of length t on a compact, negatively curved surface in the limit of large t. If the initial direction vector of the geodesic is chosen according to the Liouville measure, then it is not difficult to show that the number N(t) of self-intersections by time t grows like κt2, where κ = κM is a positive constant depend...

A recent paper (Pitteri and Zanzotto, 1998) has proposed a framework for the study of the ‘arithmetic symmetry’ of multilattices (discrete triply periodic point sets in the affine space). The classical approach to multilattice symmetry considers the well known ‘space groups’, that is, the groups of affine isometries leaving a multilattice invariant. The ensuing classification counts 219 affine ...

Non-uniform hypergraphs appear in various domains of computer science as in the satisfiability problems and in data analysis. We analyse a general model where the probability for an edge of size t to belong to the hypergraph depends of a parameter ωt of the model. It is a natural generalization of the models of graphs used by Flajolet, Knuth and Pittel [1] and Janson, Knuth, Luczak and Pittel [...