Published in McSorley, J.P. (2008). Single-change circular covering designs. Discrete Mathematics, 197/198, 561-588


A single-change circular covering design (scccd) based on the set [v] = {1, . . . ,v} with block size k is an ordered collection of b blocks, B = {B1, . . . ,Bb}, each Bi ⊂ [v], which obey: (1) each block differs from the previous block by a single element, as does the last from the first, and, (2) every pair of [v] is covered by some Bi. The object is to minimize b for a fixed v and k. We present some minimal constructions of scccds for arbitrary v when k = 2 and 3, and for arbitrary k when k+1 ≤ v ≤ 2k. Tight designs are those in which each pair is covered exactly once. Start-Finish arrays are used to construct tight designs when v > 2k; there are 2 non-isomorphic tight designs with (v, k) = (9, 4), and 12 with (v, k) = (10, 4). Some non-existence results for tight designs, and standardized, element-regular, perfect, and column-regular designs are also considered.