Embedding Hamiltonian paths in faulty arrangement graphs with the backtracking method

The arrangement graph, denoted by A_n,k, is a generalization of the star graph. A recent work by S.Y. Hsieh et al. (1999) showed that when n-k⩾4 and k=2 or n-k⩾4+[k/2] and k⩾3, A_n,k with k(n-k)-2 random edge faults, can embed a Hamiltonian cycle. In this paper, we generalize Hsieh et al. work by embedding a Hamiltonian path between arbitrary two distinct vertices of the same A_n,k. To overcome the difficulty arising from random selection of the two end vertices, a new embedding method, based on a backtracking technique, is proposed. Our results can tolerate more edge faults than Hsieh et al. results as k⩾7 and 7⩽n-k⩽3+[k/2], although embedding a Hamiltonian path between arbitrary two distinct vertices is more difficult than embedding a Hamiltonian cycle