Embedding longest fault-free paths onto star graphs with more vertex faults

Abstract

The n-dimensional star graph S n belongs to a class of bipartite graphs, and it is recognized as an attractive alternative to the hypercube. Since S 1 , S 2 , and S 3 have trivial structures, we focus our attention on S n with n ⩾ 4 in this paper. Let F ( S n ) be the set of vertex faults. Previously, it was shown that when | F ( S n ) | ⩽ n - 5 , S n with n ⩾ 6 can embed a longest fault-free path of length at least n ! - 2 | F ( S n ) | - 1 (respectively, n ! - 2 | F ( S n ) | - 2 ) between two arbitrary vertices in different partite sets (respectively, the same partite set) [Longest fault-free paths in star graphs with vertex faults, Theoretical Computer Science 262 (2001) 215–227]. In this paper, we improve the above result by tolerating more faults ( | F ( S n ) | ⩽ n - 3 ) to embed a longest fault-free path between two arbitrary vertices in S n , n ⩾ 4 .