Colored matroid

{{Short description|Abstract structure with colored elements}} In mathematics, a '''colored matroid''' is a matroid 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,<ref>{{citation | last = Zaslavsky | first = Thomas | doi = 10.2307/2153985 | issue = 1 | journal = Transactions of the American Mathematical Society | mr = 1080738 | pages = 317–347 | title = Strong Tutte functions of matroids and graphs | volume = 334 | year = 1992| jstor = 2153985 | doi-access = free }}.</ref> which generalizes the Tutte polynomial of a signed graph of {{harvtxt|Kauffman|1989}}.<ref>{{citation | last = Kauffman | first = Louis H. | doi = 10.1016/0166-218X(89)90049-8 | issue = 1–2 | journal = Discrete Applied Mathematics | mr = 1031266 | pages = 105–127 | title = A Tutte polynomial for signed graphs | volume = 25 | year = 1989| doi-access = free | citeseerx = 10.1.1.183.2851 }}.</ref>

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.<ref>{{citation | last1 = Maffioli | first1 = Francesco | last2 = Rizzi | first2 = Romeo | last3 = Benati | first3 = Stefano | doi = 10.1016/j.dam.2007.04.015 | issue = 15 | journal = Discrete Applied Mathematics | mr = 2351979 | pages = 1958–1970 | title = Least and most colored bases | volume = 155 | year = 2007| doi-access = free }}.</ref>

==References== {{reflist}}

Category:Matroid theory