Group divisible designs with block sizes from K_(1(3)) and Kirkman frames of type h~um~1

摘要

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~um~1 with K_(1(3)) = {k: k ≡ 1 mod 3} and Kirkman frames of type h~um~1. First, we determine completely the spectrum for uniform K_(1(3))-GDDs of type gu. Then, we consider the entire existence problem for non-uniform K_(1(3))-GDDs of type g~um~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~um~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~um~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~um~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.