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 k-cycle system of order v with index A, denoted by CS（v, k, λ）, is a collection A of k-cycles （blocks） of Kv such that each edge in Kv appears in exactly λ blocks of A. A large set of CS（v, k, λ）s is a partition of the set of all k-cycles of Kv into CS（v, k, λ）s, and is denoted by LCS（v, k, λ）. A （v - 1）-cycle in K, is called almost Hamilton. The completion of the existence problem for LCS（v, v- 1,λ） depends only on one case： all v ≥ 4 for λ=2. In this paper, it is shown that there exists an LCS（v,v - 1,2） for all v ≡ 2 （mod 4）, v ≥ 6.
