In this paper, we investigate the existence of incomplete group divisible designs （IGDDs） with block size four, group-type （g, h） u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg（u 1） ≡ 0 （mod 3）, λ（g h）（u 1） ≡ 0 （mod 3）, and λu（u 1）（g 2 h 2 ） ≡ 0 （mod 12）. These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved.
A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） （X,B） is called pure and denoted by PMTS（v） if （x, y, z） ∈ B implies （z, y, x） ∈B. A large set of MTS（v）s （LMTS（v）） is a collection of v - 2 pairwise disjoint MTS（v）s on a v-set. A self-converse large set of PMTS（v）s, denoted by LPMTS＊ （v）, is an LMTS（v） containing [ v-2/2] converse pairs of PMTS（v）s. In this paper, some results about the existence and non-existence for LPMTS＊ （v） are obtained.
