  In this note we construct two infinite snark families which have high oddness
and low circumference compared to the number of vertices. Using this
construction, we also give a counterexample to a suggested strengthening of
Fulkerson's conjecture by showing that the Petersen graph is not the only
cyclically 4-edge connected cubic graph which require at least five perfect
matchings to cover its edges. Furthermore the counterexample presented has the
interesting property that no 2-factor can be part of a cycle double cover.
