AI 中文总结
本文研究Johnson图在一般边故障、匹配故障和顶点故障下的哈密顿连通性，证明其达到最优容错界限，并给出构造性路由算法，仿真验证了其有效性。
AI 中文摘要
Johnson图 $J(n,k)$ 是一类经典的高度对称网络，已知在无故障情况下是哈密顿连通的。本文研究了在三种故障模型下（即一般边故障、匹配故障和顶点故障）的哈密顿连通性。对于一般边故障，我们证明了当 $n\geq4$ 时，删除任意至多 $k(n-k)-3$ 条边后，$J(n,k)$ 仍保持哈密顿连通。由于 $J(n,k)$ 是 $k(n-k)$-正则的，这达到了基于度数的自然上界。接着我们考虑匹配故障，该模型排除了多个故障链路集中于同一顶点的情况，允许更大的故障集。我们证明当 $n\geq5$ 时，删除任意匹配（包括存在时的完美匹配）后，$J(n,k)$ 仍保持哈密顿连通。对于顶点故障，我们证明当 $n\geq5$ 时，$J(n,k)$ 是 $(n-2)$-顶点故障容错的哈密顿连通图。所有三个结果都是构造性的，并引出了递归的容错哈密顿路由算法。对多达 $12{,}870$ 个顶点的 Johnson 图的仿真结果表明，路由算法成功为所有测试的源-目的对构造了无故障哈密顿路径，实测执行时间随网络规模近似线性增长。这些结果为不同故障模式下的 Johnson 图建立了一个统一的容错哈密顿连通性框架。
英文摘要
Johnson graphs $J(n,k)$ are a classical family of highly symmetric networks known to be Hamiltonian-connected in the fault-free setting. In this paper, we investigate their Hamiltonian connectivity under three failure models, namely general edge faults, matching faults, and vertex faults. For general edge faults, we prove that $J(n,k)$ remains Hamiltonian-connected after the deletion of any set of at most $k(n-k)-3$ edges for $n\geq4$. Since $J(n,k)$ is $k(n-k)$-regular, this attains the natural degree-based upper bound for Hamiltonian connectivity. We then consider matching faults, which exclude the concentration of multiple faulty links at a single vertex and permit substantially larger fault sets. We show that $J(n,k)$ remains Hamiltonian-connected after the deletion of an arbitrary matching for $n\geq5$, including a perfect matching whenever one exists. For vertex failures, we prove that $J(n,k)$ is $(n-2)$-vertex-fault-tolerant Hamiltonian-connected for $n\geq5$. All three results are constructive and lead to recursive fault-tolerant Hamiltonian routing algorithms. Simulation results on Johnson graphs with up to $12{,}870$ vertices further show that the routing algorithms successfully construct fault-free Hamiltonian paths for all tested source-destination pairs, with measured execution times exhibiting near-linear growth with network size. These results establish a unified fault-tolerant Hamiltonian-connectivity framework for Johnson graphs under different failure patterns.
