发表机构
York University（约克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究哈密顿路径重构图$R(G)$的连通性，证明当$G$包含哈密顿路径的平方作为生成子图时$R(G)$连通且可用$O(n^2)$次回咬操作重构，并构造了高连通但重构图不连通的例子。
AI 中文摘要
图 $G$ 的一条哈密顿路径上的一个回咬操作，从路径的一个端点 $u$ 添加 $G$ 的一条边到顶点 $v$，并删除路径在 $v$ 处的一条边，使得结果仍然是哈密顿路径。我们研究重构图 $R(G)$ 的连通性，其顶点是 $G$ 的哈密顿路径，当单个回咬操作将一个路径转换为另一个路径时，两个顶点相邻。我们证明，只要 $G$ 包含哈密顿路径的平方作为生成子图，$R(G)$ 就是连通的，并且任意两条哈密顿路径可以使用 $O(n^2)$ 次回咬操作进行重构。将该定理与关于生成平方路径的已知结果相结合，可得出对于满足 $\delta(G)\ge\lceil\frac{2n-1}3\rceil$ 的每个图，对于满足 $a,2a\in S$（其中 $a$ 与 $n$ 互素）的循环图 $\mathrm{Circ}(n;S)$，以及对于随机图 $G(n,p)$（其中 $p\ge c/\sqrt n$，$c$ 为足够大的常数），重构图都是连通的（后者以高概率成立）。在另一个方向上，具有任意高连通性的图可能具有不连通的重构图。对于每个 $n\ge8$ 和每个 $1\le m\le\lfloor n/2\rfloor-3$，我们构造一个具有 $\kappa=\lambda=\delta=m$ 的 $n$ 个顶点的图，其重构图是不连通的，且至少有 $(n-2m-4)!$ 个连通分量。我们猜想 $\delta(G)\ge n/2$ 足以保证连通性。
英文摘要
A backbite on a Hamiltonian path of a graph $G$ adds an edge of $G$ from an endpoint $u$ of the path to a vertex $v$ and deletes an edge of the path at $v$ so that the result is again a Hamiltonian path. We study the connectivity of the reconfiguration graph $R(G)$, whose vertices are the Hamiltonian paths of $G$, with two adjacent when a single backbite carries one to the other. We prove that $R(G)$ is connected whenever $G$ contains the square of a Hamiltonian path as a spanning subgraph, and that any two Hamiltonian paths can be reconfigured using $O(n^2)$ backbites. Combining the theorem with known results on spanning squared paths gives connectivity for every graph with $δ(G)\ge\lceil\frac{2n-1}3\rceil$, for the circulants $\mathrm{Circ}(n;S)$ with $a,2a\in S$ for some $a$ coprime to $n$, and, with high probability, for the random graph $G(n,p)$ with $p\ge c/\sqrt n$, for a sufficiently large constant $c$. In the other direction, graphs with arbitrarily high connectivity may have disconnected reconfiguration graphs. For every $n\ge8$ and every $1\le m\le\lfloor n/2\rfloor-3$ we construct a graph on $n$ vertices with $κ=λ=δ=m$ whose reconfiguration graph is disconnected, with at least $(n-2m-4)!$ components. We conjecture that $δ(G)\ge n/2$ suffices for connectivity.
