The Tutte polynomial and its applications. (English) Zbl 0769.05026

Brylawski, Thomas; Oxley, James

Matroid applications, Encycl. Math. Appl. 40, 123-225 (1992).

[For the entire collection see Zbl 0742.00052.]
A function \(f\) on the class of all matroids is an isomorphism invariant if \(f(M)=f(N)\) whenever \(M\cong N\). For every element \(e\) of \(M\) define \(f(M)=f(M\backslash e)+f(M/e)\) if \(e\) is neither a loop nor an isthmus (where \(\backslash\) and / stand for deletion and for contraction, respectively) and \(f(M)=f(M(e))f(M\backslash e)\) otherwise. Such a function \(f\) is a Tutte-Grothendieck invariant.
Such and more general invariants are presented with relations to graph colouring, flows and coding theory.

Reviewer: A.Recski (Budapest)

MSC:

05B35

Combinatorial aspects of matroids and geometric lattices

Keywords:

Tutte polynomial; matroids; Tutte-Grothendieck invariant; colouring; flows; coding