Single-change circular covering designs

A single-change circular covering design (scccd) based on the set [v] = { 1 . . . . . t:} with block size k is an ordered collection o f b blocks, ~ = {Bi . . . . . B/,}, each B~ C [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 block. The object is to minimize b for a fixed v and k. We present some minimal constructions of scccds for arbitrary t: when k 2 and 3, and for arbitrary k when k + 1 ~2k; there are 2 non-isomorphic tight designs with (v, k ) (9, 4), and 12 with (v, k ) ( 1 0 , 4). Some non-existence results for tight designs, and standardized, element-regular, perfect, and column-regular designs are considered. @ 1999 Elsevier Science B.V. All rights reserved Ko'words." Single-change; Circular; Covering; Design; Tight 1. Definitions; notation; examples A single-change circular cover&9 design based on the set [v] {1 . . . . . v} with block size k is an ordered collect ion o f b blocks, .~ = {B1 . . . . . B~,}, each an unordered subset o f k distinct e lements from [v], which obey: (1) each block differs from the previous block by a single element, i.e., IBi l~qB, .1 = k 1 for i = 2 . . . . . b ; and the last block, B/,, differs from the first, BI, by a single element, i.e., IB/, f~Bll = k 1; (2) every (unordered) pair {x,y} of [v], with x ¢ y, can be writ ten as {ei,z} where eiCB~\Bi i and zEBi for some i = 2 . . . . . b, or as {e l ,z} where el ~B~\B/, and zCB~. * E-mail: [email protected]. / 0012-365X/99,$ see front matter @ 1999 Elsevier Science B.V. All rights reserved PII: S0012-365X(98)00263-5 562 J.P. McSorley l Discrete Mathematics 197/198 (1999) 561-588 element b locks introduced pairs covered Bi ei {ei, z}, z ~ Bi B1 6 4 2 6 {6, 4} {6, 2} B2 6 3 2 3 {3,6} {3,2} Ba 6 3 5 5 {5,6} {5,3} B4 631 1 {1,6} {1,3} B5 431 4 {4, 3} {4, 1} B6 451 5 {5, 4} {5, 1} B7 251 2 {2, 5} {2,1} Bs 241 4 {4, 2} {4,1} £, an economical scecd(6, 3, 8) 3)3, a tight 123 2 3 4 3 4 5 451 5 1 2