  In contrast to matroids, vf-safe delta-matroids have three kinds of minors
and are closed under the operations of twist and loop complementation. We show
that the delta-matroids representable over GF(4) with respect to the nontrivial
automorphism of GF(4) form a subclass of the vf-safe delta-matroids closed
under twist and loop complementation. In particular, quaternary matroids are
vf-safe.
  Using this result, we show that the matroid of a bicycle space of a
quaternary matroid M is obtained from M by using loop complementation. As a
consequence, the matroid of a bicycle space of a quaternary matroid M is
independent of the chosen representation. This also leads to, e.g., an
extension of a known parity-type characterization of the bicycle dimension, a
generalization of the tripartition of Rosenstiehl and Read [Ann. Disc. Math.
(1978)], and a suitable generalization of the dual notions of bipartite and
Eulerian binary matroids to a vf-safe delta-matroids.
  Finally, we generalize a number of results concerning the Penrose polynomial
from binary matroids to vf-safe delta-matroids. In this general setting the
Penrose polynomial turns out to have a recursive relation much like the
recursive relation of the Tutte polynomial.

Combinatorics