Group divisible designs with block sizes from and Kirkman frames of type

Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for K 1 ( 3 ) -GDDs of type g u m 1 with K 1 ( 3 ) = { k : k ≡ 1 mod 3 } and Kirkman frames of type h u m 1 . First, we determine completely the spectrum for uniform K 1 ( 3 ) -GDDs of type g u . Then, we consider the entire existence problem for non-uniform K 1 ( 3 ) -GDDs of type g u m 1 with m > 0 . We show that, for each given g , up to a small number of undetermined cases of u , the necessary conditions on ( u , m ) for the existence of a K 1 ( 3 ) -GDD of type g u m 1 are also sufficient, except possibly when u ≡ 2 mod 4 for g ≡ 3 mod 6 and when u ≡ 6 mod 12 for g ≡ 1 , 5 mod 6 . Finally, a similar result for Kirkman frames of type h u m 1 is obtained. We show that, for each given h , up to a small number of undetermined cases of u , the necessary conditions on ( u , m ) for the existence of a Kirkman frame of type h u m 1 are also sufficient, except possibly when u ≡ 2 mod 4 for h ≡ 6 mod 12 and when u ≡ 6 mod 12 for h ≡ 2 , 10 mod 12 .