In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a colored matroid is a
matroid
In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms ...
whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first ''n'' positive integers, or the sign set .
The interest in colored matroids is through their invariants, especially the colored
Tutte polynomial
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contai ...
, which generalizes the Tutte polynomial of a
signed graph
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.
A signed graph is balanced if the product of edge signs around every cycle is positive. The name "signed graph" and the no ...
of .
There has also been study of optimization problems on matroids where the objective function of the optimization depends on the set of colors chosen as part of a matroid basis.
[.]
See also
*
Bipartite matroid
*
Rota's basis conjecture
In linear algebra and matroid, matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of Basis (linear algebra), bases, named after Gian-Carlo Rota. It states that, if ''X'' is either a vector space of dimension ...
References
Matroid theory
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