Rainbow and monochromatic circuits and cuts in binary matroids
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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 M either contains a rainbow colored cut or a monochromatic circuit as well. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids. Motivated by a conjecture of Bérczi 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.
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