H-designs with the properties of resolvability or (1,2)-resolvability

An H-design is said to be (1, alpha)-resolvable, if its block set can be partitioned into alpha-parallel classes, each of which contains every point of the design exactly alpha times. When alpha = 1, a (1, alpha)-resolvable H-design of type g (n) is simply called a resolvable H-design and denoted by RH(g (n) ), for which the general existence problem has been determined leaving mainly the case of g a parts per thousand 0 (mod 12) open. When alpha = 2, a (1, 2)-RH(1 (n) ) is usually called a (1, 2)-resolvable Steiner quadruple system of order n, for which the existence problem is far from complete. In this paper, we consider these two outstanding problems. First, we prove that an RH(12 (n) ) exists for all n a parts per thousand yen 4 with a small number of possible exceptions. Next, we give a near complete solution to the existence problem of (1, 2)-resolvable H-designs with group size 2. As a consequence, we obtain a near complete solution to the above two open problems.