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,{{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

}}. which generalizes the Tutte polynomial of a signed graph of {{harvtxt|Kauffman|1989}}.{{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

}}.

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.{{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

}}.