FAULT-TOLERANT HYPERCUBES WITH SMALL DEGREE

For a given N-vertex graph H, a graph G obtained from H by adding t ve rtices and some edges is called a t-FT (t-fault-tolerant) graph for H if even after deleting any t vertices from G, the remaining graph cont ains H as a subgraph. For the n-dimensional cube Q(n) with N vertices, a t-FT graph with an optimal number O(tN + t(2)) of added edges and m aximum degree of O(N + t), and a t-FT graph with O(tN log N) added edg es and maximum degree of O(tlogN) have been known. In this paper, we i ntroduce some t-FT graphs for Q(n) with an optimal number O(tN + t(2)) of added edges and small maximum degree. In particular, we show a t-F T graph for Q(n) with 2ctN + ct(2) (log N/c)(c) added edges and maximu m degree of O(N/log(c/2) N) + 4ct.