Fault-tolerant edge and vertex pancyclicity in alternating group graphs

摘要

In [J.-M. Chang, J.-S. Yang. Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760–767] the authors claim that every alternating group graph AGn is (n − 4)-fault-tolerant edge 4-pancyclic. Which means that if the number of faults ∣F∣ ⩽ n − 4, then every edge in AGn − F is contained in a cycle of length ℓ, for every 4 ⩽ ℓ ⩽ n!/2 − ∣F∣. They also claim that AGn is (n − 3)-fault-tolerant vertex pancyclic. Which means that if ∣F∣ ⩽ n − 3, then every vertex in AGn − F is contained in a cycle of length ℓ, for every 3 ⩽ ℓ ⩽  n!/2 − ∣F∣. Their proofs are not complete. They left a few important things unexplained. In this paper we fulfill these gaps and present another proofs that AGn is (n − 4)-fault-tolerant edge 4-pancyclic and (n − 3)-fault-tolerant vertex pancyclic.