Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v, 4, 1) RBIBDs. The problem we investigate is the existence of (v, 4, 1) RBIBDs containing a (w, 4, 1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w). The necessary conditions for the existence of an IRB(v,w) are that v 4w and v ≡ w ≡ 4 (mod 12). We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever w ≡ 16 (mod 60) and whenever w 1852. We also give some results on pairwise balanced designs on v points containing (at least one) block of sizew, i.e., a (v, {K,w∗}, 1)PBD. If the list of permitted block sizes, K5, contains all integers of size 5 or more, and v,w ∈ K5, then a necessary condition on this PBD is v 4w + 1. We show this condition is not sufficient for any w 5 and give the complete spectrum (in v) for 5 w 8, as well as showing the condition v 5w is sufficient with some definite exceptions for w = 5 and 6, and some possible exceptions when w = 15, namely 77 v 79. The existence of this PBD implies the existence of an IRB(12v + 4, 12w + 4). If the list of permitted block sizes,K1(4), contains all integers ≡ 1 (mod 4), and v,w ∈ K1(4), then a necessary condition on this PBD is v 4w + 1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w 1037. The existence of this PBD implies the existence of an IRB(3v + 1, 3w + 1). © 2007 Elsevier B.V. All rights reserved.