\({\mathrm{TS}}(v,\lambda )\) with Cyclic 2-Intersecting Gray Codes: \(v\equiv 0\) or \(4\pmod {12}\)

A \({\mathrm{TS}}(v,{\lambda })\) is a pair \((V,\mathcal {B})\) where V contains v points and \(\mathcal {B}\) contains 3-element subsets of V so that each pair in V appears in exactly \({\lambda }\) blocks. A 2-block intersection graph (2-BIG) of a \({\mathrm{TS}}(v,{\lambda })\) is a graph where each vertex is represented by a block from the \({\mathrm{TS}}(v,{\lambda })\) and each pair of blocks \(B_i,B_j\in \mathcal {B}\) are joined by an edge if \(|B_i\cap B_j|=2\). We show that there exists a \({\mathrm{TS}}(v,{\lambda })\) for \(v\equiv 0\) or \(4\pmod {12}\) whose 2-BIG is Hamiltonian for all admissible \((v,{\lambda })\). This is equivalent to the existence of a \({\mathrm{TS}}(v,{\lambda })\) with a cyclic 2-intersecting Gray code.