Single change neighbor designs

A singl~ change neighbor design SCN(v, k) is a way of selecting a sequence of cycles of length k from a complete graph on v vertices with the properties: any cycle is derived from the preceding one by changing one vertex; and every· edge is covered in at least one cycle. Some properties of these designs are investigated. In particular, designs with the smallest possible number of cycles are constructed for all v when k 4. We shall define a general block design (V, B) with parameters (v, k) to consist of a v-set V together with a family B of subsets of size k (called blocks) of V. A covering design of index .\ is a block design such that, given two members x and y of V, there are at least .\ blocks in B which contain {x,y}. A general discussion of is found in [1]. Single-change covering designs are discussed, for example in [3]. A single-change covering design SC(v, k) consists of a covering design of index 1 with parameters (v,k), together with an ordering B = (B 1 ,B 21 • • ,Bb) of the block-set B, such that for 1 ::; i < b, Bi+l \Bi has exactly one element. For practical purposes it is desirable to minimize the number b of blocks in an SC(v, k), which we shall call