Adjacent Vertices Fault Tolerance Hamiltonian Laceability of Star Graphs

Let Sn be an n-dimensional Star graph. In this paper, we show that Sn − F is Hamiltonian laceable where F is the set of f ≤ (n − 4) pairs of adjacent faulty vertices, Sn − F is Hamiltonian where F is the set of f ≤ (n − 3) pairs of adjacent faulty vertices. We also show that Sn − F is hyper-Hamiltonian laceable where F is the set of f ≤ (n − 4) pairs of adjacent faulty vertices. Applying these results, we also construct the fault-free cycle with length n!− 2f ′ +2 in Sn −F ′ where F ′ is the faulty vertices set with at least a black vertex and a white vertex for |F | = f ′ ≤ n− 3.