Arithmetic progressions of cycles in outer-planar graphs

AbstractA question of Erdős asks if every graph with minimum degree 3 must contain a pair of cycles whose lengths differ by 1 or 2. Some recent work of Häggkvist and Scott (see Arithmetic progressions of cycles in graphs, preprint), whilst proving this, also shows that minimum degree 500k2 guarantees the existence of cycles whose lengths are m,m+2,m+4,…,m+2k for some m—an arithmetic progression of cycles. In like vein, we prove that an outer-planar graph of order n, with bounded internal face size, and outer face a cycle, must contain a sequence of cycles whose lengths form an arithmetic progression of length exp((clogn)1/3−loglogn). Using this we give an answer for outer-planar graphs to a question of Erdős concerning the number of different sets which can be achieved as cycle spectra

Completing some Partial Latin Squares

AbstractWe show that any partial 3 r× 3 r Latin square whose filled cells lie in two disjoint r×r sub-squares can be completed. We do this by proving the more general result that any partial 3 r by 3 r Latin square, with filled cells in the top left 2r× 2 r square, for which there is a pairing of the columns so that in each row there is a filled cell in at most one of each matched pair of columns, can be completed if and only if there is some way to fill the cells of the top left 2 r× 2 r square

Culture and university in the late Middle Ages: a recent Italian study and its European dimensions

This short essay considers Paolo Rosso’s book as a contribution to the history of medieval universities; it discusses the feasibility and merits of works of synthesis by single authors, and reflects on the Italian perspective it offers and the European dimensions of the book

Dialectic approach to multidisciplinary practice in requirements engineering

This thesis develops an approach that supports multidisciplinary practice in requirements engineering. It is argued that multidisciplinary requirements engineering practice is ineffective, and some specific problems for multidisciplinary practice are identified. It is also suggested that the incommensurability of conflicting paradigms is an underlying cause of the problems in multidisciplinary practice, and a number of criteria for support to overcome such problems are proposed. A form of methodological support, which it is claimed may help overcome some of the problems associated with multidisciplinary practice in requirements engineering, is developed. This methodological support takes the form of a dialectic process, and its associated products, which is conceptualised and then operationalised. As an illustration of the methodological support offered to multidisciplinary practice, the operationalisation of the dialectic process is applied to requirements constructed by the use of two different requirements engineering techniques from two different disciplines (representing two different paradigms), in the domain of Accident and Emergency healthcare. Finally, the application of the operationalisation of the dialectic process is assessed with respect to the criteria for support for multidisciplinary practice proposed earlier, and this assessment is used to reconceptualise the dialectic process. The limitations of the research are identified, and possibilities for future work proposed. This thesis is aimed primarily at the requirements engineering community, and in particular the practising requirements engineer. It makes two contributions to knowledge supporting the practices of requirements engineering. First, the thesis contributes two types of substantive discipline knowledge: an explanation of why multidisciplinary practice in requirements engineering is problematic; and the proposal of criteria for support to allay the difficulties of multidisciplinary practice. It is suggested that these criteria might be used in the development of new types of support to overcomes such difficulties, or in the assessmment of new requirements engineering techniques that claim to address multidisciplinary practice. Second, the thesis contributes methodological knowledge in the form of a dialectic approach that offers a new way of reasoning about requirements engineering. This methodological knowledge takes two forms: a generic dialectic approach that might be applied by requirements engineering practitioners to requirements, generated by a wide range of requirements engineering techniques, representing alternative paradigms; and a specific instantiation of the dialectic approach using the MUSE method and the Grounded Theory method, that might be used in its current form by requirements engineering practitioners to support their own multidisciplinary practice