We present a linear-time algorithm that, given as input (i) a bipartite Pfaffian graph $G$ of minimum degree three, (ii) a Hamiltonian cycle $H$ in $G$, and (iii) an edge $e$ in $H$, outputs at least three other Hamiltonian cycles through the edge $e$ in $G$. This linear-time complexity of finding another Hamiltonian cycle given one is in sharp contrast to the problem of deciding the existence of a Hamiltonian cycle, which is NP-complete already for cubic bipartite planar graphs; such graphs are Pfaffian. Also, without the degree requirement, we show that it is NP-hard to find another Hamiltonian cycle in a bipartite Pfaffian graph. We present further improved algorithms for finding optimal traveling salesperson tours and counting Hamiltonian cycles in bipartite planar graphs with running times that are not known to hold in general planar graphs. We prove our results by a new structural technique that efficiently witnesses each Hamiltonian cycle $H$ through an arbitrary fixed anchor edge $e$ in a bipartite Pfaffian graph using a two-coloring of the vertices as advice that is unique to $H$. Previous techniques -- the Cut&Count technique of Cygan et al. [FOCS'11, TALG'22] in particular -- were able to reduce the Hamiltonian cycle problem only to essentially counting problems; our results show that counting can be avoided by leveraging properties of bipartite Pfaffian graphs. Our technique also has purely graph-theoretical consequences; for example, we show that every cubic bipartite Pfaffian graph has either zero or at least six distinct Hamiltonian cycles; the latter case is tight for the cube graph.

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