The Fa,b,c conjecture is true, II

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. (c) 2006 Elsevier Inc. All rights reserved.

Minimal Homogeneous Latin Trades

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d >= 3, and m >= 1.75d(2) + 3. We also improve this bound for small values of d. Our proof relies on the construction of cyclic sequences whose adjacent sums are distinct. (c) 2006 Elsevier B.V. All rights reserved.

Computing with elation groups

We have proved that elation groups of a certain infinite family of Roman generalized quadrangles are not isomorphic to those of associated flock generalized quadrangles. The proof is theoretical, but is based upon detailed computations. Here we elaborate on the explicit computer calculations which inspired the proof.