Group divisible designs with block sizes from K1(3)K1(3) and Kirkman frames of type hum1hum1

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)K1(3)-GDDs of type gum1gum1 with K1(3)={k:k≡1mod3}K1(3)={k:k≡1mod3} and Kirkman frames of type hum1hum1. First, we determine completely the spectrum for uniform K1(3)K1(3)-GDDs of type gugu. Then, we consider the entire existence problem for non-uniform K1(3)K1(3)-GDDs of type gum1gum1 with m>0m>0. We show that, for each given gg, up to a small number of undetermined cases of uu, the necessary conditions on (u,m)(u,m) for the existence of a K1(3)K1(3)-GDD of type gum1gum1 are also sufficient, except possibly when u≡2mod4u≡2mod4 for g≡3mod6g≡3mod6 and when u≡6mod12u≡6mod12 for g≡1,5mod6g≡1,5mod6. Finally, a similar result for Kirkman frames of type hum1hum1 is obtained. We show that, for each given hh, up to a small number of undetermined cases of uu, the necessary conditions on (u,m)(u,m) for the existence of a Kirkman frame of type hum1hum1 are also sufficient, except possibly when u≡2mod4u≡2mod4 for h≡6mod12h≡6mod12 and when u≡6mod12u≡6mod12 for h≡2,10mod12h≡2,10mod12.