Ring embedding in faulty star graphs

Abstract

In this paper, we consider the ring embedding problem in faulty star graphs. Our embedding is based on the path transition scheme and node borrow technique in the ring of 4-dimensional substars with evenly distributed faults. Let S-n be the n-dimensional star graph having n! nodes. We will show that a ring of length n! - 2f can be found in Sn when the number of faulty nodes f is at most n - 3. In the worst case, the loss of 2f nodes in the size of fault-free ring is inevitable because the star graph is bipartite. In addition, this result is superior to the best previous result [15] that constructs the ring of length n! - 4f under the same fault condition. Moreover, by extending this result into the star graph with both node and edge faults simultaneously, we can find the fault-free ring of length n! - 2f, in Sn when it contains f(n) faulty nodes and f(e) faulty edges such that f(n) + f(e) less than or equal to n - 3.