\(G\)-Designs Having a Prescribed Number of Blocks in Common

Abstract

Two \(G\)-designs \((X, \mathcal{A}_1)\) and \((X, \mathcal{A}_2)\) are said to intersect in \(m\) blocks if \(|\mathcal{A}_1 \cap \mathcal{A}_2| = m\). In this paper, we complete the solution of the intersection problem for \(G\)-designs, where \(G\) is a connected graph of size five which contains a cycle.