Round-dance neighbour designs from terraces

Pairwise Display of High-Dimensional Information via Eulerian Tours and Hamiltonian Decompositions

“…This method has the added advantage that for n = 2m, the Eulerian is composed of m Hamiltonian paths, while for odd n = 2m+1, it is composed of m Hamiltonian cycles. The constructions given here have been attributed to Walecki by Lucas (1892), and are sometimes described as Lucas-Walecki constructions (Bailey et al, 2003). A disadvantage is that these necessarily generate only a single class of isomorphic decompositions which, though potentially huge, cannot include those produced from possibly thousands of other Hamiltonian decomposition classes that are non-isomorphic to this one.…”

“…Bailey et al (2003) call these decompositions round-dance neighbour designs They relate them to Latin and Tuscan squares and give a number of other constructions.…”

Decomposing Complete 3-Uniform Hypergraph Kn(3) into 7-cycles

“…In fact, two different definit ions of a Hamiltonian chain and Hamiltonian cycle are the same. A decomposition of the complete k -uniform hpergraphs into Hamiltonian cycles has been considered in [7] to [14]. In paper [7], Hamiltonian decompositions of K can be decomposed into 7-cycles.…”

A decomposition of (λK υ)+ with extended triangles

“…His seminal work has resulted in a wealth of mathematical literature in both pure mathematics, as discussed in this article, and applied mathematics, such as the work on neighbor designs and on optimal networks (see [1], [10], [7], for example). For this, and for his devotion to developing both this captivating area of mathematics and young mathematicians (count the second author among those in this second category -at least one time young!)…”

“…Such a quasigroup is clearly idempotent (by (1)) and symmetric (by (2)), but also satisfies the identity…”