Search results for "Discrete Mathematics"

Central Units, Class Sums and Characters of the Symmetric Group

2010

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S n in characteristic zero, and we show that they are units in very special instances.

On the decision problem for the guarded fragment with transitivity

2002

The guarded fragment with transitive guards, [GF+TG], is an extension of GF in which certain relations are required to be transitive, transitive predicate letters appear only in guards of the quantifiers and the equality symbol may appear everywhere. We prove that the decision problem for [GF+TG] is decidable. This answers the question posed in (Ganzinger et al., 1999). Moreover, we show that the problem is 2EXPTIME-complete. This result is optimal since the satisfiability problem for GF is 2EXPTIME-complete (Gradel, 1999). We also show that the satisfiability problem for two-variable [GF+TG] is NEXPTIME-hard in contrast to GF with bounded number of variables for which the satisfiability pr…

On the Finite Satisfiability Problem for the Guarded Fragment with Transitivity

2005

We study the finite satisfiability problem for the guarded fragment with transitivity. We prove that in case of one transitive predicate the problem is decidable and its complexity is the same as the general satisfiability problem, i.e. 2Exptime-complete. We also show that finite models for sentences of GF with more transitive predicate letters used only in guards have essentially different properties than infinite ones.

Optical Routing of Uniform Instances in Cayley Graphs

2001

Abstract Abstract We consider the problem of routing uniform communication instances in Cayley graphs. Such instances consist of all pairs of nodes whose distance is included in a specified set U. We give bounds on the load induced by these instances on the links and for the wavelength assignment problem as well. For some classes of Cayley graphs that have special symmetry property (rotational graphs), we are able to construct routings for uniform instances such that the load is the same for each link of the graph.

On the Low-Dimensional Steiner Minimum Tree Problem in Hamming Metric

2011

It is known that the d-dimensional Steiner Minimum Tree Problem in Hamming metric is NP-complete if d is considered to be a part of the input. On the other hand, it was an open question whether the problem is also NP-complete in fixed dimensions. In this paper we answer this question by showing that the problem is NP-complete for any dimension strictly greater than 2. We also show that the Steiner ratio is 2 - 2/d for d ≥ 2. Using this result, we tailor the analysis of the so-called k-LCA approximation algorithm and show improved approximation guarantees for the special cases d = 3 and d = 4.

Covering and differentiation

1995

ℓ-distant Hamiltonian walks in Cartesian product graphs

2009

Abstract We introduce and study a generalisation of Hamiltonian cycles: an l-distant Hamiltonian walk in a graph G of order n is a cyclic ordering of its vertices in which consecutive vertices are at distance l. Conditions for a Cartesian product graph to possess such an l-distant Hamiltonian walk are given and more specific results are presented concerning toroidal grids.

A Loopless Generation of Bitstrings without p Consecutive Ones

2001

Let F n (p) be the set of all n-length bitstrings such that there are no p consecutive ls. F n (p) is counted with the pth order Fibonacci numbers and it may be regarded as the subsets of {1, 2,…, n} without p consecutive elements and bitstrings in F n (p) code a particular class of trees or compositions of an integer. In this paper we give a Gray code for F n (p) which can be implemented in a recursive generating algorithm, and finally in a loopless generating algorithm.

On Fine and Wilf's theorem for bidimensional words

2003

AbstractGeneralizations of Fine and Wilf's Periodicity Theorem are obtained for the case of bidimensional words using geometric arguments. The domains considered constitute a large class of convex subsets of R2 which include most parallelograms. A complete discussion is provided for the parallelogram case.

On the Deskins index complex of a maximal subgroup of a finite group

1999

AbstractLet M be a maximal subgroup of a finite group G. A subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G is contained in M. The set, I(M), of all completions of M is called the index complex of M in G. Set P(M) = {C ϵ I(M) ¦ C} is maximal in I(M) and G = CM. The purpose of this note is to prove: A finite group G is solvable if and only if, for each maximal subgroup M of G, P(M) contains element C with CK(C) nilpotent.