On (K4,K4 - e)-designs

In [5], the first author posed the problem of determining the spectrum of (K4,K4 - e)-designs. In this article, we solve this problem, and also determine the spectrum of (K4, K4 - e)-designs with exactly one K4 (or, equivalently, the spectrum of (K4 - e)-designs with a hole of size 4). We also improve the bound for embedding a partial S(2,4,υ) into a (K4, K4 - e)-design given in [5].