Partitioning sets of triples into small planes

We study partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2, 2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions) or copies of some planes of each type (mixed partitions). We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in several cases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We construct such partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, and an affine partition for v = 18. Using these as starter partitions, we prove that Fano partitions exist for v = 7(n) + 1, 13(n) + 1, 27(n) + 1, and affine partitions for v = 8(n) + 1, 9(n) + 1, 17(n) + 1. In particular, both Fano and affine partitions exist for v = 3(6n) + 1. Using properties of 3-wise balanced designs, we extend these results to show that affine partitions also exist for v = 3(2n). Similarly, mixed partitions are shown to exist for v = 8(n), 9(n), 11(n) + 1.

A subset-oriented algorithm for minimizing the number of steps required for synthesis of cyclic-peptide libraries

Libraries of cyclic peptides are being synthesized using combinatorial chemistry for high throughput screening in the drug discovery process. This paper describes the min_syn_steps.cpp program (available at http://www.imb.uq.edu.au/groups/smythe/tran), which after inputting a list of cyclic peptides to be synthesized, removes cyclic redundant sequences and calculates synthetic strategies which minimize the synthetic steps as well as the reagent requirements. The synthetic steps and reagent requirements could be minimized by finding common subsets within the sequences for block synthesis. Since a brute-force approach to search for optimum synthetic strategies is impractically large, a subset-orientated approach is utilized here to limit the size of the search. (C) 2002 Elsevier Science Ltd. All rights reserved.

On the volume of 4-cycle trades

A 4-cycle trade of volume t corresponds to a simple graph G without isolated vertices, where the edge set can be partitioned into t 4-cycles in at least two different ways such that the two collections of 4-cycles have no 4-cycles in common. The foundation of the trade is v = \V(G)\. This paper determines for which values oft and a there exists a 4-cycle trade of volume t and foundation v.

The spectrum of minimal defining sets of some Steiner systems

For a design D, define spec(D) = {\M\ \ M is a minimal defining set of D} to be the spectrum of minimal defining sets of D. In this note we give bounds on the size of an element in spec(D) when D is a Steiner system. We also show that the spectrum of minimal defining sets of the Steiner triple system given by the points and lines of PG(3,2) equals {16,17,18,19,20,21,22}, and point out some open questions concerning the Steiner triple systems associated with PG(n, 2) in general. (C) 2002 Elsevier Science B.V. All rights reserved.

More on exact bicoverings of 12 points

The minimum number of incomplete blocks required to cover, exactly lambda times, all t-element subsets from a set V of cardinality v (v > t) is denoted by y(lambda, t; v). The value of g(2, 2; v) is known for v = 3, 4,..., 11. It was previously known that 14 less than or equal to g(2, 2; 12) less than or equal to 16. We prove that g(2, 2; 12) greater than or equal to 15.

Factorizations of the complete graph into C-5-factors and 1-factors

In this paper, we show that K-10n can be factored into alpha C-5-factors and beta 1-factors for all non-negative integers alpha and beta satisfying 2alpha + beta = 10(n) - 1.

Skolem-type difference sets for cycle systems

Cyclic m-cycle systems of order v are constructed for all m greater than or equal to 3, and all v = 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle system of K-v - F for all m greater than or equal to 3, and all v = 2(mod 2m).

Decompositions of complete graphs into theta graphs with fewer than ten edges

A theta graph is a graph consisting of three pairwise internally disjoint paths with common end points. Methods for decomposing the complete graph K-nu into theta graphs with fewer than ten edges are given.