In
mathematics, a colored matroid is a
matroid
In combinatorics, a branch of mathematics, 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 axiomatically, the most significant being ...
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 contain ...
, 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
In mathematics, a bipartite matroid is a matroid all of whose circuits have even size.
Example
A uniform matroid U^r_n is bipartite if and only if r is an odd number, because the circuits in such a matroid have size r+1.
Relation to bipartite gra ...
*
Rota's basis conjecture
References
Matroid theory
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