A note on the 2-Factor Hamiltonicity Conjecture

The 2-factor Hamiltonicity Conjecture by Funk, Jackson, Labbate, and Sheehan [JCTB, 2003] asserts that all cubic, bipartite graphs in which all 2-factors are Hamiltonian cycles can be built using a simple operation starting from $K_{3,3}$ and the Heawood graph. We discuss the link between this conjecture and matching theory, in particular by showing that this conjecture is equivalent to the statement that the two exceptional graphs in the conjecture are the only cubic braces in which all 2-factors are Hamiltonian cycles, where braces are connected, bipartite graphs in which every matching of size at most two is contained in a perfect matching. In the context of matching theory this conjecture is especially noteworthy as $K_{3,3}$ and the Heawood graph are both strongly tied to the important class of Pfaffian graphs, with $K_{3,3}$ being the canonical non-Pfaffian graph and the Heawood graph being one of the most noteworthy Pfaffian graphs. Our main contribution is a proof that the Heawood graph is the only Pfaffian, cubic brace in which all 2-factors are Hamiltonian cycles. This is shown by establishing that, aside from the Heawood graph, all Pfaffian braces contain a cycle of length four, which may be of independent interest.