On \((K_4, K_4 – e)\)-Designs

Abstract

In \([5]\), the first author posed the problem of determining the spectrum of \((K_4, K_4 – e)\)-designs. In this article, we solve this problem, and also determine the spectrum of \((K_4, K_4 – e)\)-designs with exactly one \(K_4\) (or, equivalently, the spectrum of \((K_4 – e)\)-designs with a hole of size \(4\)). We also improve the bound for embedding a partial \(S(2,4,v)\) into a \((K_4, K_4 – e)\)-design given in \([5]\).