Rainbow and monochromatic circuits and cuts in binary matroids

TL;DR

This paper investigates the interplay between rainbow circuits and monochromatic cuts in binary matroids, providing new characterizations, bounds, and conditions related to colorings, rigidity, and matroid structure.

Contribution

It introduces novel results linking rainbow circuits and monochromatic cuts in binary matroids, characterizes binary matroids via reductions, and explores coloring constraints in relation to rigidity.

Findings

Binary matroids with r colors contain either a rainbow circuit or a monochromatic cut.

Characterization of binary matroids through reductions to partition matroids.

Existence of rainbow circuit-free colorings with limited color repetitions in certain graphs.

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.