  The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By
exploiting our recent results on resistance of snarks, we construct infinite
classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with
circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and
$0.990$, respectively. In contrast, the dominating cycle conjecture implies
that the circumference ratio of a cyclically $4$-edge-connected cubic graph is
at least $0.75$.
  In addition, we construct snarks with large girth and large circumference
deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On
stable cycles and cycle double covers of graphs with large circumference, Disc.
Math. 312 (2012), 2540--2544].
