Strong Tutte Functions of Matroids and Graphs

A strong Tutte function of matroids is a function of finite matroids which satisfies F ( M 1$M2) = F ( M 1 ) F ( M 2 ) and F ( M ) = aeF(M\e) + b e F ( M / e ) for e not a loop or coloop of M ,where ae , be are scalar parameters depending only on e . We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial. The classification remains valid if the domain is any minor-closed class of matroids containing all three-point matroids. Similar classifications hold for strong Tutte functions of colored matroids, where the parameters depend on the color of e ,and for strong Tutte functions of graphs and edge-colored graphs whose values do not depend on the attachments of loops. The latter classification implies new characterizations of Kauffman's bracket polynomials of signed graphs and link diagrams. A Tutte-Grothendieck invariant of matroids is a function F from (finite) matroids to a domain of scalars, which satisfies the multiplicative, additive, and invariance laws: (A) F( M ) = F(M\e) + F(M/e) if e is a nonseparating point of M (that is, neither a loop nor coloop), and Brylawski in [ l ] proved all such functions to be the evaluations of a certain two-variable polynomial function of matroids known as the Tutte polynomial, tM(x ,y ) , or equivalently (as Crapo had shown in [2]) of the rank polynomial (or "rank-generating polynomial") RM(u , v) ,whose definition is very different but which equals tM(u + 1 , v + 1 ) . These results were extensions to matroids of seminal ideas introduced originally for graphs by Tutte [ l 1, 121. Research into polynomial invariants of knots by Thistlethwaite (especially [9]) and Kauffman led the latter to define a version of the Tutte polynomial for Received by the editors August 5, 1990. Presented to the American Mathematical Society at the 93rd Summer Meeting in Columbus, Ohio, on August 11, 1990. 1991 Mathematics Subject Classification. Primary 05B35; Secondary 05C99, 57M25.