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.

Dartboard arrangements with a concave penalty function

This paper investigates combinatorial arrangements of the dartboard to maximize a penalty function derived from the differences of adjacent sectors. The particular penalty function is constructed by summing the absolute differences of neighbouring sectors raised to a power between zero and one. The arrangement to give the maximum penalty is found