An optimal result on fault-tolerant cycle-embedding in alternating group graphs

Abstract

The alternating group graph was proposed as an interconnection network topology for computing systems. It has many advantages over n-cubes and star graphs. Let F be a set of faulty vertices in an n-dimensional alternating group graph AG n . A cycle of length ℓ is said to be an ℓ-cycle. A previous result in [J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760–767] showed that, for n ⩾ 4 , AG n − F is pancyclic (i.e., AG n − F contains an ℓ-cycle for each ℓ with 3 ⩽ ℓ ⩽ | V ( AG n − F ) | ) if | F | ⩽ n − 2 . Although the fault-tolerances of AG 4 are tight, the results are not optimal for n ⩾ 5 . In this paper, for the same problem, we give an optimal result: for n ⩾ 4 , AG n − F is pancyclic if | F | ⩽ 2 n − 6 . Since a graph G is pancyclic, then clearly it is hamiltonian. Thus, for n ⩾ 4 , AG n − F is also hamiltonian if | F | ⩽ 2 n − 6 . This result is also an improvement over the previous ones given in [J.-M. Chang, J.-S. Yang, Y.-L. Wang, Y. Cheng, Panconnectivity, fault-tolerant hamiltonicity and hamiltonian-connectivity in alternating group graphs, Networks 44 (2004) 302–310].