  In 1971, Tutte wrote in an article that "it is tempting to conjecture that
every 3-connected bipartite cubic graph is hamiltonian". Motivated by this
remark, Horton constructed a counterexample on 96 vertices. In a sequence of
articles by different authors several smaller counterexamples were presented.
The smallest of these graphs is a graph on 50 vertices which was discovered
independently by Georges and Kelmans. In this article we show that there is no
smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic
graphs in the literature have cyclic 4-cuts -- even if they have girth 6 -- it
is natural to ask whether this is a necessary prerequisite. In this article we
answer this question in the negative and give a construction of an infinite
family of non-hamiltonian cyclically 5-connected bipartite cubic graphs.
  In 1969, Barnette gave a weaker version of the conjecture stating that
3-connected planar bipartite cubic graphs are hamiltonian. We show that
Barnette's conjecture is true up to at least 90 vertices. We also report that a
search of small non-hamiltonian 3-connected bipartite cubic graphs did not find
any with genus less than 4.
