  Given a matroid together with a coloring of its ground set, a subset of its
elements is called rainbow colored if no two of its elements have the same
color. We show that if a binary matroid of rank $r$ is colored with exactly $r$
colors, then $M$ either contains a rainbow colored circuit or a monochromatic
cut. As the class of binary matroids is closed under taking duals, this
immediately implies that if $M$ is colored with exactly $n-r$ colors, then $M$
either contains a rainbow colored cut or a monochromatic circuit. As a
byproduct, we give a characterization of binary matroids in terms of reductions
to partition matroids.
  Motivated by a conjecture of B\'erczi et al., we also analyze the relation
between the covering number of a binary matroid and the maximum number of
colors or the maximum size of a color class in any of its rainbow circuit-free
colorings. For simple graphic matroids, we show that there exists a rainbow
circuit-free coloring that uses each color at most twice only if the graph is
$(2,3)$-sparse, that is, it is independent in the $2$-dimensional rigidity
matroid. Furthermore, we give a complete characterization of minimally rigid
graphs admitting such a coloring.
