Longest paths and cycles in faulty star graphs

Abstract

In this paper, we investigate the star graph S n 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 S n , n ⩾ 4 , there is a fault-free path of length at least n ! - 2 f v - 1 , and there is a path of length at least n ! - 2 f v - 2 joining a pair of vertices with the same color, when the number of faulty elements is n - 3 or less. Here, f v is the number of faulty vertices. S n , n ⩾ 4 , with at most n - 2 faulty elements has a fault-free cycle of length at least n ! - 2 f v 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 S n , 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.