The Hamilton–Waterloo problem: the case of Hamilton cycles and triangle-factors

AbstractWe discuss a special case of the Hamilton–Waterloo problem in which a 2-factorization of Kn is sought consisting of 2-factors of two kinds: Hamiltonian cycles, and triangle-factors. We determine completely the spectrum of solutions for several infinite classes of orders n

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 the Hamilton-Waterloo problem for bipartite 2-factors

Given two 2-regular graphs F1 and F2, both of order n, the Hamilton-Waterloo Problem for F1 and F2 asks for a factorization of the complete graph Kn into a1 copies of F1, a2 copies of F2, and a 1-factor if n is even, for all nonnegative integers a1 and a2 satisfying a1+a2=?n-12?. We settle the Hamilton-Waterloo Problem for all bipartite 2-regular graphs F1 and F2 where F1 can be obtained from F2 by replacing each cycle with a bipartite 2-regular graph of the same order