The Gordon game

In 1992, about 30 years after Gordon introduced group sequencings to construct row-complete Latin squares, John Isbell introduced the idea of competitive sequencing, the Gordon Game. Isbell investigated the Gordon Game and found solutions for groups of small order. The purpose of this thesis is to analyze the Gordon Game and develop a brute force method of determining solutions to the game for all groups of order 12 (up to isomorphism) as well as for abelian groups of order less than 20. The method used will be a depth first search program written in MATLAB. Consequently, group representation using matrices will be studied within the thesis --Document

Sectionable terraces and the (generalised) Oberwolfach problem

AbstractThe generalised Oberwolfach problem requires v people to sit at s round tables of sizes l1,l2,…,ls (where l1+l2+⋯+ls=v) for successive meals in such a way that each pair of people are neighbours exactly λ times. The problem is denoted OP(λ;l1,l2,…,ls) and if λ=1, which is the original problem, this is abbreviated to OP(l1,l2,…,ls). It was known in 1892, though different terminology was then used, that a directed terrace with a symmetric sequencing for the cyclic group of order 2n can be used to solve OP(2n+1). We show how terraces with special properties can be used to solve OP(2;l1,l2) and OP(l1,l1,l2) for a wide selection of values of l1, l2 and v. We also give a new solution to OP(2;l,l) that is based on Z2l−1. Solutions to the problem are also of use in the design of experiments, where solutions for tables of equal size are called resolvable balanced circuit Rees neighbour designs

On some sequencing problems in finite groups

AbstractA finite group is called Z-sequenceable if its non-identity elements can be listed x1, x2, …, xn so that xixi+1 for i = 1, 2, …, n − 1. Various necessary and sufficient conditions are determined for such sequencings to exist. In particular, it is proved that if n ⩾ 3, then the symmetric group Sn is not Z-sequenceable

Circular external difference families, graceful labellings and cyclotomy

(Strong) circular external difference families (which we denote as CEDFs and SCEDFs) can be used to construct nonmalleable threshold schemes. They are a variation of (strong) external difference families, which have been extensively studied in recent years. We provide a variety of constructions for CEDFs based on graceful labellings ($\alpha$-valuations) of lexicographic products $C_n \boldsymbol{\cdot} K_{\ell}^c$, where $C_n$ denotes a cycle of length $n$. SCEDFs having more than two subsets do not exist. However, we can construct close approximations (more specifically, certain types of circular algebraic manipulation detection (AMD) codes) using the theory of cyclotomic numbers in finite fields

Incremental Lower Bounds for Additive Cost Planning Problems

We present a novel method for computing increasing lower bounds on the cost of solving planning problems, based on repeatedly solving and strengthening the delete relaxation of the problem. Strengthening is done by compiling select conjunctions into new atoms, similar to the P*m construction. Because it does not rely on search in the state space, this method does not suffer some of the weaknesses of admissible search algorithms and therefore is able to prove higher lower bounds for many problems that are too hard for optimal planners to solve, thus narrowing the gap between lower bound and cost of the best known plan, providing better assurances of plan quality

Constructing R-sequencings and terraces for groups of even order

The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R*-sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order 8t for t congruent to 1, 2, 3 or 4 (mod7). As much is known about which odd-order abelian groups are R*-sequenceable, we have constructions of R-sequencings for many new families of abelian groups. The construction is generalisable in several directions, leading to a wide array of new R-sequenceable and terraceable non-abelian groups of even order