Rainbow and monochromatic circuits and cocircuits in binary matroids

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 an n-element rank r binary matroid M is colored with exactly r colors, then M either contains a rainbow colored circuit or a monochromatic cocircuit. 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 cocircuit or a monochromatic circuit. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids.& nbsp;Motivated by a conjecture of Berczi, Schwarcz and Yamaguchi, 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.(C) 2022 The Author(s). Published by Elsevier B.V.& nbsp;