Fault-tolerant hypercubes with small degree

For a given N-vertex graph H, a graph G obtained from H by adding t vertices 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 contains H as a subgraph. For an N-vertex hypercube Q(N), a t-FT graph with an optimal number O(tN + t(2)) of added edges and maximum degree of O(N + t), and a t-FT graph with O(tN log N) added edges and maximum degree of O(t log N) have been known. In this paper, we introduce 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-FT graph for Q(N) with 2ct N + ct(2) (log N/c)(c) added edges and maxmum degree of O(n/log(c/2) N) + 4ct.