Embedding cycles of various lengths into star graphs with both edge and vertex faults

Abstract

The n-dimensional star graph S n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F v and F e be the sets of faulty vertices and faulty edges of S n , respectively. We prove that S n − F v − F e contains a fault-free cycle of every even length from 6 to n! − 2∣ F v ∣ with ∣ F v ∣ + ∣ F e ∣ ⩽ n − 3 for every n ⩾ 4. We also show that S n − F v − F e contains a fault-free path of length n! − 2∣ F v ∣ − 1 (respectively, n! − 2∣ F v ∣ − 2) between two arbitrary vertices of S n in different partite sets (respectively, the same partite set) with ∣ F v ∣ + ∣ F e ∣ ⩽ n − 3 for every n ⩾ 4.