Minimum Embedding of a \(K_3\)-Design into a Balanced Incomplete Block Design of Index \(\lambda \geq 2\)

Abstract

Let \(H\) be a subgraph of \(G\). An \(H\)-design \((V, \mathcal{C})\) of order \(v\) and index \(\lambda\) is embedded into a \(G\)-design \((X, \mathcal{B})\) of order \(v+w\), \(w \geq 0\), and index \(\lambda\), if \(\mu \leq \lambda\), \(V \subseteq X\) and there is an injective mapping \(f: \mathcal{C} \rightarrow \mathcal{B}\) such that \(B\) is a subgraph of \(f(B)\) for every \(B \in \mathcal{C}\).
For every pair of positive integers \(v\) and \(\lambda\), we determine the minimum value of \(w\) such that there exists a balanced incomplete block design of order \(v+w\), index \(\lambda \geq 2\) and block-size \(4\) which embeds a \(K_3\)-design of order \(v\) and index \(\mu = 1\).