Embed a Longest Path between Arbitrary Two Vertices of a Faulty Star Graph

The star graph Sn has been recognized as an attractive alternative to the hypercube. Since S1, S2, and S3 have trivial structures, we focus our attention on Sn with n≥4 in this paper. Let Fv denote the set of faulty vertices in Sn. We show that when |Fv|≤n−5, Sn with n≥6 can embed a fault−free path of length n!-2|Fv|−2 (n!-2|Fv|−1, respectively) between arbitrary two vertices of even (odd, respectively) distance. Since Sn is bipartite with two partite sets of equal size, the embedded path is the longest in the worst case. Besides, we also show that when |Fv|=n−4 or n−3, Sn with n≥4 can embed a fault−free path of length at least n!-4|Fv|−10 (n!-4|Fv|−9, respectively) between arbitrary two vertices of even (odd, respectively) distance. Since Sn is regular of degree n-1, |Fv|=n−3 is maximum in the worst case in order to embed a longest fault−free path between arbitrary two vertices of Sn.