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Given a matroid together with a coloring of its ground set, a subset of itselements is called rainbow colored if no two of its elements have the samecolor. 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 monochromaticcut. As the class of binary matroids is closed under taking duals, thisimmediately implies that if $M$ is colored with exactly $n-r$ colors, then $M$either contains a rainbow colored cut or a monochromatic circuit. As abyproduct, we give a characterization of binary matroids in terms of reductionsto partition matroids.  Motivated by a conjecture of B\'erczi et al., we also analyze the relationbetween the covering number of a binary matroid and the maximum number ofcolors or the maximum size of a color class in any of its rainbow circuit-freecolorings. For simple graphic matroids, we show that there exists a rainbowcircuit-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 rigiditymatroid. Furthermore, we give a complete characterization of minimally rigidgraphs admitting such a coloring.

Rainbow and monochromatic circuits and cuts in binary matroids