Fault-tolerant Hamiltonian connectivity of Johnson graphs

Abstract

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.