Existence of doubly near resolvable (v, 4, 3) BIBDs

Doubly near resolvable m-cycle systems

An m-cycle system C of λKn is said to be near resolvable if the mcycles in C can be partitioned into near parallel classes R1, R2, . . . , Rλn/2, each one of which is a 2-factor of λKn − v for some vertex v in λKn. If an NR(n,m, λ)-CS has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted by DNR(n,m, λ)-CS. For m = 2, 3, DNR(n,m, 2)-CSs are known as Room frames ...

The Existence of DNR ( mt + 1 , m , m − 1 ) - BIBDs

A (v, m, m − 1)-BIBD D is said to be near resolvable (NRBIBD) if the blocks of D can be partitioned into classes R1, R2, . . . , Rv such that for each point x of D, there is precisely one class having no block containing x and each class contains precisely v − 1 points of the design. If an (v, m, λ)-NRBIBD has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted D...

The Existence of Kirkman Squares-Doubly Resolvable (v, 3, 1)-BIBDs

A Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v; μ, λ), is a t × t array (t = λ(v − 1)/μ(k − 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V , and (3) the collection of blocks obtained from the non-empty cells of the array is a (...

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...

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