The spectrum of α-resolvable λ-fold (K_4 - e)-designs

Abstract

A λ-fold G-design is said to be α-resolvable if its blocks can be partitioned into classes such that every class contains each vertex exactly α times. In this paper we study the α-resolvability for λ-fold (K 4 − e)-designs and prove that the necessary conditions for their existence are also sufficient, without any exception.

Construction of S(3) (2, 3)-Designs of Any Index

“…An H (h) -design, or also a design of type H (h) or a system of H (h) , with order v and index λ, is a pair Σ = (X, B), where X is a finite set of cardinality v, whose elements are called vertices, and B is a collection of hypergraphs over X, called blocks, all isomorphic to H (h) , under the condition that every h-subset of X is a hyperedge of exactly λ hypergraphs of the collection B. An H (h) -design, of order v and index λ, is also called an H (h) -decomposition of λK (h) v (see, for example, [1][2][3][4][5]). It is important to note that in the definition of index λ, we do not require that the blocks be distinct.…”