A Subset Expansion of the Coloured Tutte Polynomial

Bollobás and Riordan introduce a Tutte polynomial for coloured graphs and matroids in [3]. We observe that this polynomial has an expansion as a sum indexed by the subsets of the ground-set of a coloured matroid, generalizing the subset expansion of the Tutte polynomial. We also discuss similar expansions of other contraction–deletion invariants of graphs and matroids. 1. The coloured Tutte polynomial The Tutte polynomial or dichromate is an invariant of graphs and matroids which has been found to be associated with a variety of important and seemingly unrelated matters – vertex colourings, acyclic orientations and flows in graphs, reliability of communication networks, statistical mechanics and knot theory. In several of these instances it is natural to modify the Tutte polynomial to reflect weights associated with the matroid elements or graph edges being considered; for instance, consider the reliability of a network whose edges function with various probabilities, or an invariant of a knot diagram in which it is necessary to distinguish overpassing arcs from underpassing arcs. In [3] Bollobás and Riordan construct a 'universal' Tutte polynomial of weighted and coloured graphs, and mention that the invariant is easily extended to matroids. The purpose of this note is to observe that part of the theory of the ordinary (unweighted) Tutte polynomial generalizes to their universal invariant. Although we can direct the reader to several sources of background information – [1, 7] for graph theory, [5, 9] for matroid theory, and [1, 7, 8, 10] for the Tutte polynomial – [3] is our primary reference. For the sake of convenience, we discuss matroids rather than graphs in this section; in Section 2 below we briefly discuss other contraction–deletion invariants, including non-matroidal invariants of graphs. Suppose (M, c) is a coloured matroid, i.e., a matroid M on a finite set E together with a function c mapping E into some set Λ of colours.