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.

翻译：我们提出了一种线性时间算法，对于输入：(i) 最小度为3的双部Pfaffian图$G$，(ii) $G$中的一个哈密顿环$H$，(iii) $H$中的一条边$e$，该算法输出$G$中通过边$e$的至少三个其他哈密顿环。给定一个哈密顿环后找到另一个哈密顿环的线性时间复杂度，与判断哈密顿环存在性（该问题对三次双部平面图已是NP完全的，而这类图正是Pfaffian图）形成鲜明对比。此外，若无度条件限制，我们证明在双部Pfaffian图中找到另一个哈密顿环是NP困难的。我们进一步改进了在双部平面图中寻找最优旅行商回路和计数哈密顿环的算法，其运行时间在一般平面图中尚不成立。我们通过一种新的结构技术证明结果，该技术利用顶点二染色作为对每个哈密顿环$H$唯一的建议，高效地验证双部Pfaffian图中通过任意固定锚边$e$的每个哈密顿环$H$。以往技术（尤以Cygan等人的Cut&Count技术 [FOCS'11, TALG'22]）只能将哈密顿环问题归结为本质上的计数问题；我们的结果表明，通过利用双部Pfaffian图的特性可避免计数。该技术还带来纯粹图论意义上的结果，例如：每个三次双部Pfaffian图要么无哈密顿环，要么至少存在六个不同的哈密顿环，后者对于立方体图是紧的。