Two types of switchable λλ-fold (K4−e)(K4−e)-designs

In this paper the existence of two types of edge-switchable (K4−e,λ)(K4−e,λ)-designs are investigated. It is established that the necessary conditions for a Type I switchable (λKv,K4−e)(λKv,K4−e)-design is λ(v−1)≡0(mod5), v≥4v≥4, and these conditions are also sufficient except for (v,λ)=(11,1)(v,λ)=(11,1). The necessary conditions for a Type II switchable (λKv,K4−e)(λKv,K4−e)-design are λv(v−1)≡0(mod10), v≥4v≥4, and these conditions are also sufficient with a small number of possible exceptions.