Rainbow and monochromatic circuits and cocircuits in binary matroids

Bérczi, Kristóf; Schwarcz, Tamás

doi:10.1016/j.disc.2022.112830

Cited by 3 publications

(2 citation statements)

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“…A milestone result of this area is Edmonds' arborescences theorem [11], characterizing the existence of k pairwise disjoint arborescences in a directed graph. Bérczi and Schwarcz [5] studied rainbow circuit-free colorings of binary matroids in general, and showed 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. Such a coloring can be identified with a so-called reduction to a partition matroid, which is closely related to the problem of packing rainbow spanning trees.…”

Section: Previous Work Bérczi and Schwarczmentioning

confidence: 99%

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On the complexity of packing rainbow spanning trees

Bérczi¹,

Csáji²,

Király³

2022

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One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. Bérczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest.In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spanning trees, an extension of the problem of packing arborescences in directed graphs which was answered by Edmonds' seminal result on disjoint arborescences. We complement his result by showing that it is NP-complete to decide whether an edgecolored graph contains two disjoint rainbow spanning trees. Our complexity result holds even for the very special case when the graph is the union of two spanning trees and each color class contains exactly two edges. As a corollary, we give a negative answer to a question on the decomposition of oriented k-partition-connected digraphs.

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Section: Previous Work Bérczi and Schwarczmentioning

confidence: 99%

“…Despite impressive achievements, the complexity of finding disjoint rainbow spanning trees remained an intriguing open question that was raised by many, see e.g. [5,24].…”

Section: Previous Work Bérczi and Schwarczmentioning

confidence: 99%