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In this paper, we prove that cyclic hamiltonian cycle systems of the complete graph minus a 1-factor, K n − I, exist if and only if n ≡ 2, 4(mod 8) and n = 2p α with p prime and α ≥ 1.

In this paper, we settle Alspach's problem in the case of Hamilton cycles and 5-cycles; that is, we show that for all odd integers n ≥ 5 and all nonnegative integers h and t with hn + 5t = n(n − 1)/2, the complete graph K n decomposes into h Hamilton cycles and t 5-cycles and for all even integers n ≥ 6 and all nonnegative integers h and t with hn + 5t =… (More)

Let G of order n be the vertex-disjoint union of an even and an odd cycle. It is known that there exists a G-decomposition of K v for all v ≡ 1 (mod 2n). We use an extension of the Bose construction for Steiner triple systems and a recent result on the Oberwolfach Problem for 2-regular graphs with two components to show that there exists a G-decomposition… (More)

A classic theorem of Veblen states that a connected graph G has a cycle decomposition if and only if G is Eulerian. The number of odd cycles in a cycle decomposition of an Eulerian graph G is therefore even if and only if G has even size. It is conjectured that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph G with m edges… (More)

- Tina Helms, Heather Jordon
- 2012

A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples 5t+1}\{k}. In this talk, we will give necessary and sufficient conditions on t and k for the existence of a k-extended Skolem-type 5-tuple difference set of order t. We also consider hooked k-extended Skolem-type 5-tuple difference sets of order t and provide necessary and… (More)

In this paper, we prove that directed cyclic hamiltonian cycle systems of the complete symmetric digraph, K * n , exist if and only if n ≡ 2 (mod 4) and n = 2p α with p prime and α ≥ 1. We also show that directed cyclic hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (K n − I) * , exist if and only… (More)

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