Group divisible designs with block sizes from K₁₍₃₎ and Kirkman frames of type h^um¹

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 K₁₍₃₎-GDDs of type g ^um¹ with K₁₍₃₎={k:k≡1mod3} and Kirkman frames of type h^um¹. First, we determine completely the spectrum for uniform K₁₍₃₎-GDDs of type g^u. Then, we consider the entire existence problem for non-uniform K₁₍₃₎-GDDs of type g^um¹ 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 K₁₍₃₎-GDD of type g^um¹ are also sufficient, except possibly when u≡2mod4 for g≡3mod6 and when u≡6mod12 for g≡1,5mod6. Finally, a similar result for Kirkman frames of type h^um¹ 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 ^um¹ are also sufficient, except possibly when u≡2mod4 for h≡6mod12 and when u≡6mod12 for h≡2,10mod12.