Fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs

The arrangement graph A_n,k is a generalization of the star graph. There are some results concerning fault Hamiltonicity and fault Hamiltonian connectivity of the arrangement graph. However, these results are restricted in some particular cases and, thus, are less completed. We improve these results and obtain a stronger and simpler statement. Let n-k≥2 and F⊆V(A_n,k)∪E(A_n,k). We prove that A_n,k-F is Hamiltonian if |F|≤k(n-k)-2 and A_n,k-F is Hamiltonian connected if |F|≤k(n-k)-3. These results are optimal.