The undirected de Bruijn graph: fault tolerance and routing algorithms

The undirected version of the de Bruijn graph, also called the shift-and-replace graph, has N=kⁿ vertices, maximum degree k, and minimum degree k-2. A.H. Esfahanian and S.L. Hakimi (1985) have shown that this graph has diameter n=log_kN, connectivity 2k-2, and has an increase in diameter of at most log_kn+4 in the presence of up to 2k-3 fault vertices. The author tightens this bound significantly. It is shown that the increase in the diameter of this graph is at most log_klog _φn+6+log_k5, where φ is the golden ratio. The methods used draw upon the theory of string overlaps and the theory of finite automata