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Softcover reprint of the hardcover 1 st edition 1999 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

No abstract

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ2) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [50] obtained the first result in this direction for the finite projective spaces).As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [11] and [20].

We discuss a wide range of matching problems in terms of a network flow model. More than this, we start up a matching theory which is very intuitive and independent from the original graph context. This first paper contains a standardized theory for the performance analysis of augmentation algorithms in a wide area of matching problems. Several optimality criteria are given which do not use cuts or barriers. As an application of our theory, the known cardinality matching algorithms of Edmonds, Kameda and Munro, and Micali and Vazirani, and the algorithm of Kocay and Stone for capacitated matching problems can be studied in their effects. From our theory a c-capacitated b-matching algorithm can be derived that behaves like the Dinic algorithm for the maximum flow problem. It will turn out that techniques for the maximum flow problem can be applied to matching problems much more explicitly than done before. A comprehensive duality theory depending on the network flow model used here will follow. Explicit algorithms for nonweighted problems will be presented in subsequent papers.

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We prove that every polarity of PG(2k − 1, q), where k ≥ 2, gives rise to a design with the same parameters and the same intersection numbers as, but not isomorphic to, PG k (2k, q). In particular, the case k = 2 yields a new family of quasi-symmetric designs. We also show that our construction provides an infinite family of counterexamples to Hamada's conjecture, for any field of prime order p. Previously, only a handful of counterexamples were known.