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