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- W4287555166 abstract "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." @default.
- W4287555166 created "2022-07-25" @default.
- W4287555166 creator A5026832894 @default.
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- W4287555166 date "2020-12-09" @default.
- W4287555166 modified "2023-09-26" @default.
- W4287555166 title "Rainbow and monochromatic circuits and cuts in binary matroids" @default.
- W4287555166 doi "https://doi.org/10.48550/arxiv.2012.05037" @default.
- W4287555166 hasPublicationYear "2020" @default.
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