  It is well-known that the circular flow number of a bridgeless cubic graph
can be computed in terms of certain partitions of its vertex-set with
prescribed properties. In the present paper, we first study some of these
properties that turn out to be useful in order to design a more efficient
algorithm for the computation of the circular flow number of a bridgeless cubic
graph. Using this algorithm, we determine the circular flow number of all
snarks up to 36 vertices as well as the circular flow number of various famous
snarks. After that, as combination of the use of our algorithm with new
theoretical results, we present an infinite family of snarks of order $8k+2$
whose circular flow numbers meet a general lower bound presented by Lukot'ka
and Skoviera in 2008. In particular, this answers a question proposed in their
paper. Moreover, we improve the best known upper bound for the circular flow
number of Goldberg snarks and we conjecture that this new upper bound is
optimal. Finally, we discuss a possible extension to the computation of the
circular flow number in the non-regular case.

Combinatorics;Discrete Mathematics