All directed BIBDs with k = 3 exist

All Directed BIBDs with k = 3 Exist

A directed BIBD with parameters (v,b,r,k,λ*) is a BIBD with parameters (v, b, r, k, 2λ*) in which each ordered pair of varieties occurs together in exactly λ* blocks. It is shown that λ*v(v 1) = 0 (mod 3) is a necessary and sufficient condition for the existence of a directed (v, b, r, k, λ*) BIBD with k = 3. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J and Skill...

Some new near resolvable BIBDs with k=7 and resolvable BIBDs with k=5

The existence of directed BIBDs

For any positive integers k¿ 3 and , let cd(k; ) denote the smallest integer such that the necessary conditions 2 (v − 1) ≡ 0 (mod k − 1) and v(v − 1) ≡ 0 (mod ( k2 )) for the existence of a DB(k; ; v) are also su3cient for every v¿ cd(k; ). In this article we provide an estimate for cd(k; ) when k ≡ 0 (mod 4) and any . Combined with the results in (Discrete Math. 222 (2000) 27–40), we complete...

A few more BIBDs with k=8 or 9

It has been shown that for k = 8 or 9, there exists a BIBD(v, k, 1) for all positive integers v == 1 or k (mod k(k-1)), with some possible exceptions. We show that such designs exist for 49 of these exceptional values. A balanced incomplete block design (BIBD) with parameters (v, k, 1) is a pair (X, A) where X is a v-set and A is a family of k-subsets (where 2 < k < v) called blocks, such that ...

Existence of near resolvable (v, k, k-1) BIBDs with k∈{9, 12, 16}

A necessary condition for existence of a (v, k, k−1) near resolvable BIBD is v ≡ 1 (mod k). In this paper, we update earlier known existence results when k ∈ {9, 12, 16}, and show this necessary condition is sufficient, except possibly for 26, 37 and 149 values of v for k = 9, 12, 16 respectively. Some new results for existence of (9, 8)-frames of type 9 are also obtained; in particular, we sho...