Gerechte Designs with Rectangular Regions

A \emph{gerechte framework} is a partition of an $n \times n$ array into $n$ regions of $n$ cells each. A \emph{realization} of a gerechte framework is a latin square of order $n$ with the property that when its cells are partitioned by the framework, each region contains exactly one copy of each symbol. A \emph{gerechte design} is a gerechte framework together with a realization. We investigate gerechte frameworks where each region is a rectangle. It seems plausible that all such frameworks have realizations, and we present some progress towards answering this question. In particular, we show that for all positive integers $s$ and $t$, any gerechte framework where each region is either an $s \times t$ rectangle or a $t\times s$ rectangle is realizable.Comment: 14 pages, 12 figure

Fighting Against Biopiracy: Does the Obligation to Disclose in Patent Applications Truly Help?

In the global fight against biopiracy, one of the key issues is to prevent the grant and exploitation of patents on traditional knowledge and genetic resources by requiring that patent applicants for inventions involving traditional knowledge and genetic resources disclose the source of those resources and provide evidence that the prior informed consent of the local owners of such resources has been obtained and that benefit-sharing agreements have been entered into with those owners. This Article argues that a legal discussion of biopiracy should analyze the obligation to disclose the use of traditional knowledge and genetic resources in an invention beyond the sanctions that are attached in case of violation of such obligations as previously discussed at the international level. These issues should be addressed in light of the key objectives to be achieved: to ensure the effective sharing of benefits resulting from the use of such resources with the local communities that own them, and to implement appropriate mechanisms for this purpose. In the course of the analysis, this Article adopts an interdisciplinary approach by referring to rules governing the legal protection of tangible and intangible cultural property in order to explore the extent to which they could be used as models for a regime of protection against the misappropriation of traditional knowledge and genetic resources. This approach is inspired by the similarity between biopiracy and the misappropriation of cultural property goods, which constitutes a kind of cultural piracy. This Article concludes that balanced, flexible, and interdisciplinary solutions are required in order to ensure that the interests of local communities are protected without unduly threatening the interests of their commercial partners

Weighted stability number of graphs and weighted satisfiability: The two facets of pseudo-Boolean optimization

We exhibit links between pseudo-Boolean optimization, graph theory and logic. We show the equivalence of maximizing a pseudo-Boolean function and finding a maximum weight stable set; symmetrically minimizing a pseudo-Boolean function is shown to be equivalent to solving a weighted satisfiability proble

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this are

Recoloring subgraphs of K2n for sports scheduling

The exploration of one-factorizations of complete graphs is the foundation of some classical sports scheduling problems. One has to traverse the landscape of such one-factorizations by moving from one of those to a so-called neighbor one-factorization. This approach amounts to modifying locally the coloring associated with a one-factorization. We consider some particular types of modifications and describe various constructions which give one-factorizations which may be modified or not by these techniques. Among those are recoloring of bichromatic cycles, altering of optimally colored subcliques of even size, or recoloring of chordless lanterns. Keywords: graph theory, one-factorization, subgraph recoloringpublishedVersio

A constrained sports scheduling problem

AbstractA real case of sports scheduling problem is presented. A calendar for two leagues has to be constructed; besides the usual restrictions on the alternation of home- and away-games, one has to consider the fact that some pairs of teams in the two leagues share the same facilities and cannot play home-games simultaneously. Furthermore depending on the results in the first games, one of the leagues is divided for the last games into two subleagues. An optimal solution is constructed by using properties of oriented factorizations of complete graphs