Finding a Hamiltonian cycle in a given graph is computationally challenging, and in general remains so even when one is further given one Hamiltonian cycle in the graph and asked to find another. In fact, no significantly faster algorithms are known for finding another Hamiltonian cycle than for finding a first one even in the setting where another Hamiltonian cycle is structurally guaranteed to exist, such as for odd-degree graphs. We identify a graph class -- the bipartite Pfaffian graphs of minimum degree three -- where it is NP-complete to decide whether a given graph in the class is Hamiltonian, but when presented with a Hamiltonian cycle as part of the input, another Hamiltonian cycle can be found efficiently. We prove that Thomason's lollipop method~[Ann.~Discrete Math.,~1978], a well-known algorithm for finding another Hamiltonian cycle, runs in a linear number of steps in cubic bipartite Pfaffian graphs. This was conjectured for cubic bipartite planar graphs by Haddadan [MSc~thesis,~Waterloo,~2015]; in contrast, examples are known of both cubic bipartite graphs and cubic planar graphs where the lollipop method takes exponential time. Beyond the lollipop method, we address a slightly more general graph class and present two algorithms, one running in linear-time and one operating in logarithmic space, that take 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 $H$, and output at least three other Hamiltonian cycles through the edge $e$ in $G$. We also 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.

翻译：在给定图中寻找哈密顿环在计算上具有挑战性，即使进一步给定图中的一个哈密顿环并要求寻找另一个，一般情况仍然困难。事实上，即便在另一个哈密顿环在结构上保证存在的情况下（如奇度图），目前已知的寻找另一个哈密顿环的算法并不比寻找第一个哈密顿环的算法显著更快。我们确定了一类图——最小度为三的二分图Pfaffian图——其中判定给定图是否为哈密顿图是NP完全的，但当输入中给定一个哈密顿环时，可以高效地找到另一个哈密顿环。我们证明了Thomason的棒棒糖方法[Ann.~Discrete Math.,~1978]（一种著名的寻找另一个哈密顿环的算法）在三次二分图Pfaffian图中以线性步数运行。这一结论由Haddadan [硕士论文, Waterloo, 2015] 针对三次二分图平面图提出猜想；相比之下，已知存在三次二分图和三次平面图的实例，其中棒棒糖方法需要指数时间。除棒棒糖方法外，我们还针对稍一般的图类提出了两种算法：一种在线性时间内运行，另一种在对数空间内运行，它们以(i)最小度为三的二分图Pfaffian图$G$、(ii)$G$中的一个哈密顿环$H$以及(iii)$H$中的一条边$e$为输入，并输出$G$中经过边$e$的至少三个其他哈密顿环。我们还提出了进一步改进的算法，用于寻找最优旅行商巡回和计数二分图平面图中的哈密顿环，其运行时间在一般平面图中尚未证明成立。