Modified frames with block size 3

Abstract

A modified (k, λ)-frame of type g u is a modified (k, λ)-GDD whose blocks can be partitioned into holey parallel classes, each of which is with respect to some group. Modified frames can be used to construct some other resolvable designs such as resolvable group divisible designs and semiframes. In this article, we shall investigate the existence of modified frames with block size 3.

On the existence of (3,λ)‐semiframes of type 3u

“…If the blocks of a ðk; lÞ-MGDD can be partitioned into parallel classes, the ðk; lÞ-MGDD is said to be resolvable and denoted by ðk; lÞ-RMGDD. A modified ðk; lÞ-frame (MF) of type g u is a ðk; lÞ-MGDD whose blocks can be partitioned into holey parallel classes, each of which is with respect to some group, see [4]. Theorem 2.4.…”

“…B is a collection of subsets of X with sizes from K, called blocks, such that every pair of points from distinct groups occurs in exactly l blocks. 4. No pair of points belonging to a group occurs in any block.…”

Almost resolvable even cycle systems of $(K_u \times K_g)(λ)$

“…RCS of (K u × K g )(λ) has been studied by many authors [1,3,4,11]. H. Cao et al [8] proved that there exists a 3 -ARCS of (K u × K g )(λ) if and only if λ(g − 1) ≡ 0(mod 2) , g(u − 1) ≡ 0(mod 3) , g ≥ 3 , and u ≥ 4 , except for (λ, g, u) = { (1,3,6), (2m + 1, 6n + 3, 6) , m ≥ 0 and n ≥ 1}. In this paper, we shall prove that the necessary conditions for a k -ARCS of (K u × K g )(λ) are also sufficient with few possible exceptions.…”