Edge colouring models for the Tutte polynomial and related graph invariants

For integer q ≥ 2, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H q , (ii) the symmetric weight enumerator for group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. We describe the general relationship between vertex and edge colouring models, deriving a result of Szegedy and generalizing a theorem of Loebl along the way. In the second half of the paper we exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q ≥ k each evaluate the number of proper edge k-colourings of G when G is Pfaffian.