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