Summary: 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 = 1-
d-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.
The moment of appearance of the first cycle in an evolving random graph has been studied
extensively in . It is known that the first cyclic component appears on average when the graph
has approximately 0.44n edges where n is the number of vertices, however, this random variable
has a huge variance, and there is a positive probability of containing a cycle when there are only
n edges, for any > 0.