The Minimality of the Georges-Kelmans Graph

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.