Fault-tolerant hypercubes with small degree

Abstract

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 (f-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/sub N/, a t-FT graph with an optimal number O(tN+t/sup 2/) of added edges and maximum degree of O(N+t), and a t-FT graph with O(fN 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/sub N/ with an optimal number O(tN+t/sup 2/) of added edges and small maximum degree. In particular, we show a t-FT graph for Q/sub N/ with 2ctN+ct/sup 2/(log N/c)/sup 3/ added edges and maximum degree of O(N/log/sup c///sup 2/N)+4ct.