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.

Decomposition of complete graphs into 5-cubes

Necessary conditions for the complete graph on n vertices to have a decomposition into 5-cubes are that 5 divides it - 1 and 80 divides it (it - 1)/2. These are known to be sufficient when n is odd. We prove them also sufficient for it even, thus completing the spectrum problem for the 5-cube and lending further weight to a long-standing conjecture of Kotzig. (c) 2005 Wiley Periodicals, Inc.

Some equitably 2-colourable cycle decompositions of multipartite graphs

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has x(i) vertices coloured c(i), i = 1, 2,..., k, and vertical bar x(i) - x(j)vertical bar

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u, w, and v are odd, ((v)(2)) - ((u)(2)) - ((w)(2)) equivalent to 0 (mod 3), and v >= w + u + max {u, w}. Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v - u - w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well-known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. (c) 2005 Wiley Periodicals, Inc.

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m is an element of {3, 5} only, and that the algebras arising from {1, 2}-perfect m-cycle systems form a variety for m is an element of {3, 5, 7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety. (c) 2006 Elsevier B.V. All rights reserved.

A flaw in the use of minimal defining sets for secret sharing schemes

It is shown that in some cases it is possible to reconstruct a block design D uniquely from incomplete knowledge of a minimal defining set for D. This surprising result has implications for the use of minimal defining sets in secret sharing schemes.

Maximal triangle trades with foundation 5 (mod 6) - the final case

The maximum possible volume of a simple, non-Steiner (v, 3, 2) trade was determined for all v by Xhosrovshahi and Torabi (Ars Combinatoria 51 (1999), 211-223), except that in the-case v equivalent to 5 (mod 6), v >= 23, they were only able to provide an upper, bound on the volume. In this paper we construct trades with volume equal to that bound for all v equivalent to 5 (mod 6), thus completing the problem.

Edge-coloured cube decompositions

An edge-colored graph is a graph H together with a function f:E(H) → C where C is a set of colors. Given an edge-colored graph H, the graph induced by the edges of color c C is denoted by H(c). Let G, H, and J be graphs and let μ be a positive integer. A (J, H, G, μ) edge-colored graph decomposition is a set S = {H 1,H 2,...,H t} of edge-colored graphs with color set C = {c 1, c 2,..., c k} such that Hi ≅ H for 1 ≤ i ≤ t; Hi (cj) ≅ G for 1 ≤ i ≤ t and ≤ j ≤ k; and for j = 1, 2,..., k, each edge of J occurs in exactly μ of the graphs H 1(c j ), H 2(c j ),..., H t (c j ). Let Q 3 denote the 3-dimensional cube. In this paper, we find necessary and sufficient conditions on n, μ and G for the existence of a (K n ,Q 3,G, μ) edge-colored graph decomposition. © Birkhäuser Verlag, Basel 2007.