Regular graphs whose subgraphs tend to be acyclic

Motivated by a problem that arises in the study of mirrored storage systems, we describe, for any fixed , > 0 and any integer d 2, explicit or randomized constructions of d-regular graphs on n > n0( , ) vertices in which a random subgraph obtained by retaining each edge, randomly and independently, with probability = 1d-1 , is acyclic with probability at least 1 - . On the other hand we show that for any d-regular graph G on n > n1( , ) vertices, a random subgraph of G obtained by retaining each edge, randomly and independently, with probability = 1+ d-1 , does contain a cycle with probability at least 1-. The proofs combine probabilistic and combinatorial arguments, with number theoretic techniques.

• Type: Journal
• Year: 2006
• Where: RSA
• Authors: Noga Alon, Eitan Bachmat
• Keywords: D-regular Graph, Random Subgraph, RSA 2006, Storage Systems