$\mathrm{T}\mathrm{S}(v,\lambda )$ with cyclic 2-intersecting Gray codes: $v\equiv 0$ or $4(\mathrm{m}\mathrm{o}\mathrm{d}12)$

A $\mathrm{T}\mathrm{S}(v,\lambda )$ 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 $\mathrm{T}\mathrm{S}(v,\lambda )$ is a graph where each vertex is represented by a block from the $\mathrm{T}\mathrm{S}(v,\lambda )$ and each pair of blocks $B_i,B_j\in \mathcal{B}$ are joined by an edge if $\mathrm{\mid }{B}_i\cap B_j\mathrm{\mid }=2$. Using constructions for $\mathrm{T}\mathrm{S}(v,\lambda )$ given by Schreiber, we show that there exists a $\mathrm{T}\mathrm{S}(v,\lambda )$ for $v\equiv 0$ or $4(\mathrm{m}\mathrm{o}\mathrm{d}12)$ whose $2$-BIG is Hamiltonian.