Longest paths and cycles in faulty star graphs

In this paper, we investigate the star graph Sn with faulty vertices and/or edges from the graph theoretic point of view. We show that between every pair of vertices with different colors in a bicoloring of Sn, n ≥ 4, there is a fault-free path of length at least n!− 2fv − 1, and there is a path of length at least n!− 2fv − 2 joining a pair of vertices with the same color, when the number of faulty elements is n − 3 or less. Here, fv is the number of faulty vertices. Sn, n ≥ 4, with at most n−2 faulty elements has a fault-free cycle of length at least n! − 2fv unless the number of faulty elements are n− 2 and all the faulty elements are edges incident to a common vertex. It is also shown that Sn, n ≥ 4, is strongly hamiltonian-laceable if the number of faulty elements is n− 3 or less and the number of faulty vertices is one or less.