Ian Anderson

Combinatorial designs : construction methods

Introduction to basics difference methods symmetric designs projective geometries orthogonal Latin squares self orthogonal Latin squares transversal designs and Wilsons theorem Steiner triple systems Kirkman triple systems Steiner systems and t-designs Room Squares and bridge tournaments balanced tournament designs whist tournaments.

Perfect matchings of a graph

Abstract Tuttes theorem on perfect matchings is considered from the viewpoint of the Marriage Problem. A short proof is exhibited and some consequences of the theorem are discussed.

Power-sequence terraces for Z n where n is an odd prime power

A power-sequence terrace for Zn is defined to be a terrace which can be partitioned into segments one of which contains merely the zero element of Zn, whilst each other segment is either (a) a sequence of successive powers of an element of Zn, or (b) such a sequence multiplied throughout by a constant. Many elegant families of such Zn terraces are constructed for values of n that are odd prime powers. The discovery of these families greatly increases the number of known constructions for terraces for Zn. Tables are provided to show clearly the constructions available for each prime power n satisfying n < 300.

New Product Theorems for Z-Cyclic Whist Tournaments

The aim of this note is to show how existing product constructions for cyclic and 1-rotational block designs can be adapted to provide a highly effective method of obtaining product theorems for whist tournaments.

Current graphs and bi‐embeddings

Let N(γ,γ′) denote the size of the smallest complete graph which cannot be edge-partitioned into two parts embeddable in closed orientable surfaces of genera γ,γ′ respectively. The theory of current graphs is used to determine the values of N(γ,γ′) in certain cases. Some related block designs are discussed.

Narcissistic half-and-half power-sequence terraces for Zn with n=pqt

Abstract A power-sequence terrace for Z n is a Z n terrace that can be partitioned into segments one of which contains merely the zero element of Z n whilst each other segment is either (a) a sequence of successive powers of an element of Z n , or (b) such a sequence multiplied throughout by a constant. If n is odd, a Z n terrace (a1,a2,…,an) is a narcissistic half-and-half terrace if ai−ai−1=an+2−i−an+1−i for i=2,3,…,(n+1)/2. Constructions are provided for narcissistic half-and-half power-sequence terraces for Z n with n=pqt where p and q are distinct odd primes and t is a positive integer. All the constructions are for terraces with as few segments as possible. Attention is restricted to constructions covering values of n with n=pqt and n

An existence theorem for cyclic triplewhist tournaments

Abstract We show that a Z-cyclic triplewhist tournament TWh(v) exists whenever v=p1α1 ⋯ prαr where the primes pi are ≡ 5 (mod 8), pi ⩾ 29. The method of construction uses the existence of a primitive root ω of each such pi (≠61) such that ω2 ±ω+1 are both squares (mod pi).

Intersection theorems and a lemma of Kleitman

Abstract A lemma of Kleitman is used to derive a simple proof of an existing theorem and to confirm part of a conjecture of Katona. The lemma is extended from subsets of a set to divisors of a number, and some new results are obtained.

On the toroidal thickness of graphs

The toroidal thickness t1(G) of a graph G is the minimum value of k such that G is the union of k graphs each of which is embeddable on a torus. We find t1(Gm), where Gm is the graph obtained from the complete graph Km by removing a Hamiltonian cycle, and we show that t1(Kn(3)) = [1/2n] for many values of n. The method of approach involves the construction of sets of triples related to Skolem triples.

Infinite families of biembedding numbers

Let N(γ, γ′) denote the size of the smallest complete graph that cannot be edge-partitioned into two parts embeddable in closed orientable sufaces of genera γ, γ′, respectively. Well-known embedding theorems are used to obtain several infinite families of values of N(γ, γ′). Some related small values of N(γ, γ′) are also discussed.