Fault-Tolerant Embedding of Longest Paths in Star Graphs with Edge Faults

In this paper, we aim to embed a longest path between every two vertices of a star graph with at most n-3 random edge faults, where n is the dimension of the star graph. Since the star graph is regular of degree n1, n-3 (edge faults) is maximum in the worst case. We show that when n≥6 and there are n-3 edge faults, the star graph can embed a longest path between every two vertices, exclusive of two exceptions in which there are at most two vertices missing from the longest path. The probabilities of the two exceptions are analyzed. When n≥6 and there are n-4 edge faults, the star graph can embed a longest path between every two vertices. The situation of n<6 is also discussed.