Group divisible designs with two associate classes, and quadratic leaves of triple systems

In this paper, we consider group divisible designs with two associate classes, completely settling the existence problem for K 3 -designs of λ 1 K n ∨ λ 2 λ 1 K m when m = 2 and when λ 1 ≥ λ 2 . We also extend a classic result of Colbourn and Rosa on quadratic leaves, finding necessary and sufficient conditions for the existence of a K 3 -decomposition of λ K n − E ( Q ) , where Q is any 2-regular subgraph of K n .