Darryn E. Bryant

3,5-Cycle decompositions

For all odd integers n and all non-negative integers r and s satisfying 3r + 5s = n(n -1)/2 it is shown that the edge set of the complete graph on n vertices can be partitioned into r 3-cycles and s 5-cycles. For all even integers n and all non-negative integers r and s satisfying 3r + 5s = n(n-2)/2 it is shown that the edge set of the complete graph on n vertices with a 1-factor removed can be partitioned into r 3-cycles and s 5-cycles

On the Hamilton-Waterloo problem

Abstract. The Hamilton-Waterloo problem asks for a 2-factorisation of Kv in which r of the 2-factors consist of cycles of lengths a1,a2,…,at and the remaining s 2-factors consist of cycles of lengths b1,b2,…,bu (where necessarily ∑i=1tai=∑j=1ubj=v). In this paper we consider the Hamilton-Waterloo problem in the case ai=m, 1≤i≤t and bj=n, 1≤j≤u. We obtain some general constructions, and apply these to obtain results for (m,n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}.

A Generalized Markov Sampler

A recent development of the Markov chain Monte Carlo (MCMC) technique is the emergence of MCMC samplers that allow transitions between different models. Such samplers make possible a range of computational tasks involving models, including model selection, model evaluation, model averaging and hypothesis testing. An example of this type of sampler is the reversible jump MCMC sampler, which is a generalization of the Metropolis–Hastings algorithm. Here, we present a new MCMC sampler of this type. The new sampler is a generalization of the Gibbs sampler, but somewhat surprisingly, it also turns out to encompass as particular cases all of the well-known MCMC samplers, including those of Metropolis, Barker, and Hastings. Moreover, the new sampler generalizes the reversible jump MCMC. It therefore appears to be a very general framework for MCMC sampling. This paper describes the new sampler and illustrates its use in three applications in Computational Biology, specifically determination of consensus sequences, phylogenetic inference and delineation of isochores via multiple change-point analysis.

Decompositions into 2-regular subgraphs and equitable partial cycle decompositions

Two theorems are proved in this paper. Firstly, it is proved that there exists a decomposition of the complete graph of order n into t edge-disjoint 2-regular subgraphs of orders m1, m2,...,mt if and only if n is odd, 3≤mi ≤ n for i = 1, 2,...,t, and m1 + m2 +...+ mt = (n 2). Secondly, it is proved that if there exists partial decomposition of the complete graph Kn of order n into t cycles of lengths m1, m2,..., mt, then there exists an equitable partial decomposition of Kn into t cycles of lengths m1, m2,..., mt. A decomposition into cycles is equitable if for any two vertices u and v, the number of cycles containing u and the number of cycles containing v differ by at most 1.

Cycle decompositions V: Complete graphs into cycles of arbitrary lengths

We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1,..., mt if and only if n is odd, 3≤mi≤n for i=1,..., t, and. We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1,..., mt if and only if n is even, 3≤mi≤n for i=1,..., t, and m1 + ··· + mt = n2- n/2.

Packing cycles in complete graphs

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.

A Family of Perfect Factorisations of Complete Bipartite Graphs

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p2 for an odd prime p. We construct a family of (p?1)/2 non-isomorphic perfect 1-factorisations of Kn, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltonian if the permutation defined by any row relative to any other row is a single cycle.

TWO-DIMENSIONAL BALANCED SAMPLING PLANS EXCLUDING CONTIGUOUS UNITS

ABSTRACT A balanced sampling plan excluding contiguous units (or BSEC for short) was first introduced by Hedayat, Rao and Stufken in 1988. These designs can be used for survey sampling when the units are arranged in one-dimensional ordering and the contiguous units in this ordering provide similar information. In this paper, we generalize the concept of a BSEC to the two-dimensional situation and give constructions of two-dimensional BSECs with block size 3. The existence problem is completely solved in the case where λ = 1.

Embeddings of m -cycle systems and incomplete m -cycle systems: m ≤14

In this paper we completely settle the embedding problem for m-cycle systems with m less than or equal to 14. We also solve the more general problem of finding m-cycle systems of K-v - K-u when m is an element of {4,6,7,8,10,12,14}.

The Doyen-Wilson theorem extended to 5-cycles

In this paper it is proved that any 5-cycle system of order u can be embedded in a 5-cycle system of order v iff v > 3u/2 and v = 1 or 5 (mod 10)