A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p 2 for an odd prime p. We construct a family of (p − 1)/2 non-isomorphic perfect 1-factorisations of K n, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamilto-nian if the permutation defined by any… (More)
MOTIVATION A consensus sequence for a family of related sequences is, as the name suggests, a sequence that captures the features common to most members of the family. Consensus sequences are important in various DNA sequencing applications and are a convenient way to characterize a family of molecules. RESULTS This paper describes a new algorithm for… (More)
The set of integers k for which there exist three latin squares of order n having precisely k cells identical, with their remaining n 2 − k cells diierent in all three latin squares, denoted by I 3 [n], is determined here for all orders n.
Cyclic m-cycle systems of order v are constructed for all m ≥ 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… (More)
A perfect 1-factorisation of a graph G is a decomposition of G into edge disjoint 1-factors such that the union of any two of the factors is a Hamiltonian cycle. Let p 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of K p+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of K p,p. If 2… (More)