A new class of critical sets in Latin squares

Previously the process of finding critical sets in Latin squares has been inside cumbersome by the complexity and number of Latin trades that, must be constructed. In this paper we develop a theory of Latin trades that yields more transparent constructions. We use these Latin trades to find a new class of critical sets for Latin squares which are a product of the Latin square of order 2 with a. back circulant Latin square of odd order.

Embedding partial G-designs where G is a 4-cycle with a pendant edge

Let D denote the graph consisting of a cycle of length 4 with a pendant edge. In this paper, two very different small embeddings of partial D-designs are presented. (c) 2005 Elsevier B.V. All rights reserved.

Packing lambda-fold complete multipartite graphs with 4-cycles

A maximum packing of any lambda-fold complete multipartite graph (where there are lambda edges between any two vertices in different parts) with edge-disjoint 4- cycles is obtained and the size of each minimum leave is given. Moreover, when lambda =2, maximum 4-cycle packings are found for all possible leaves.

Visualising energy data

The Australian energy market is in the final stages of deregulation. These changes have created a dynamic environment which is highly volatile and competitive with respect to both demand and price. Our current research seeks to visualise aspects of the National Energy Market with a view to developing techniques which may be useful in identifying significant characteristics and/or drivers of these characteristics. In order to capture the complexity of the problem we explore a suite of different visualisation techniques, which, when combined into a unified package, highlight aspects of the problem. The particular problem visualised here is "Does the date exhibit characteristics which suggest that the time of day, day of the week, or the season, aflect the variation in demand and/or price?" © Austral. Mathematical Soc. 2005.

Embedding a restricted class of partial K-4-designs

Given a partial K-4-design (X, P), if x is an element of X is a vertex which occurs in exactly one block of P, then call x a free vertex. In this paper, a technique is described for obtaining a cubic embedding of any partial K-4-design with the property that every block in the partial design contains at least two free vertices.