Generalized measures of fault tolerance for bubble sort networks

The κk and λk are two generalized measurements for fault tolerance of large-scale processing system. For the bubble sort networks Bn, this paper determines κk(Bn)=λk(Bn)=2k(n−k−1) for k ≤ n/2. The results show that to disconnect Bn with each vertex in resulting graph has at least k fault-free neighbors, at least 2k(n−k−1) faulty vertices or faulty edges have to occur. In particular, the results also settle affirmatively a conjecture proposed by Shi and Wu (Acta Math. Appl. Sin-E., 33 (4)(2017), 933–944).