Colored matroid

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,¹ which generalizes the Tutte polynomial of a signed graph of Kauffman (1989).²

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.³

References

Zaslavsky, Thomas (1992), "Strong Tutte functions of matroids and graphs", Transactions of the American Mathematical Society, 334 (1): 317–347, doi:10.2307/2153985, JSTOR 2153985, MR 1080738.
Kauffman, Louis H. (1989), "A Tutte polynomial for signed graphs", Discrete Applied Mathematics, 25 (1–2): 105–127, CiteSeerX 10.1.1.183.2851, doi:10.1016/0166-218X(89)90049-8, MR 1031266.
Maffioli, Francesco; Rizzi, Romeo; Benati, Stefano (2007), "Least and most colored bases", Discrete Applied Mathematics, 155 (15): 1958–1970, doi:10.1016/j.dam.2007.04.015, MR 2351979.

[1] Zaslavsky, Thomas (1992), "Strong Tutte functions of matroids and graphs", Transactions of the American Mathematical Society, 334 (1): 317–347, doi:10.2307/2153985, JSTOR 2153985, MR 1080738.

[2] Kauffman, Louis H. (1989), "A Tutte polynomial for signed graphs", Discrete Applied Mathematics, 25 (1–2): 105–127, CiteSeerX 10.1.1.183.2851, doi:10.1016/0166-218X(89)90049-8, MR 1031266.

[3] Maffioli, Francesco; Rizzi, Romeo; Benati, Stefano (2007), "Least and most colored bases", Discrete Applied Mathematics, 155 (15): 1958–1970, doi:10.1016/j.dam.2007.04.015, MR 2351979.

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See also

References