1-2hit |
Xuzhen XIE Takao ONO Tomio HIRATA
Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using
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 the n-dimensional cube Q(n) with N vertices, a t-FT graph with an optimal number O(tN+t2) of added edges and maximum degree of O(N+t), and a t-FT graph with O(tNlog N) added edges and maximum degree of O(tlog N) have been known. In this paper, we introduce some t-FT graphs for Q(n) with an optimal number O(tN+t2) of added edges and small maximum degree. In particular, we show a t-FT graph for Q(n) with 2ctN+ct2((logN)/C)C added edges and maximum degree of O(N/(logC/2N))+4ct.