  The 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is
the graph whose vertex set is composed of the blocks of the TTS and two
vertices are joined by an edge if the corresponding blocks intersect in exactly
two elements. The 2-BIGs are themselves interesting graphs: each component is
cubic and 3-connected, and a 2-BIG is bipartite exactly when the TTS is
decomposable to two Steiner triple systems. Any connected bipartite 2-BIG with
no Hamilton cycle is a counter-example to a conjecture posed by Tutte in 1971.
Our main result is that there exists an integer $N$ such that for all $v\geq
N$, if $v\equiv 1$ or $3\mod{6}$ then there exists a TTS($v$) whose 2-BIG is
bipartite and connected but not Hamiltonian. Furthermore, $13<N\leq 663$. Our
approach is to construct a TTS($u$) whose 2-BIG is connected bipartite and
non-Hamiltonian and embed it within a TTS($v$) where $v>2u$ in such a way that,
after a single trade, the 2-BIG of the resulting TTS($v$) is bipartite
connected and non-Hamiltonian.
