We prove that the ribbon graph polynomial of a embedded in an orientable surface is irreducible if and only neither disjoint union nor join graphs. This result analogous to fact Tutte connected non-separable.

We generalize and unify results on parametrized and coloured Tutte polynomials of graphs and matroids due to Zaslavsky, and Bollobás and Riordan. We give a generalized Zaslavsky– Bollobás–Riordan theorem that characterizes parametrized contraction–deletion functions on minor-closed classes of matroids, as well as the modifications necessary to apply the discussion to classes of graphs. In gener...

We show how the Tutte polynomial of a plane graph can be evaluated as the "homfly" polynomial of an associated oriented link. Then we discuss some consequences for the partition function of the Potts model, the Four Color Problem and the time complexity of the computation of the homfly polynomial.

We solve in the affirmative a conjecture of Brylawski, namely that the Tutte polynomial of a connected matroid is irreducible over the integers. If M is a matroid over a set E, then its Tutte polynomial is defined as T(M; x, y)= C A ı E (x − 1) r(E) − r(A) (y − 1) | A | − r(A) , where r(A) is the rank of A in M. This polynomial is an important invariant as it contains much information on the ma...

Ribbon graphs are surfaces with boundary together with a decomposition into a union of closed topological discs of two types, edges and vertices. These sets are subject to some natural axioms recalled in section 2.1. For such a generalisation of the usual graphs, B. Bollobás and O. Riordan found a topological version of the Tutte polynomial [3, 4]. In the following, we will refer to this genera...

The complexity of calculating the coefficients of the Tutte polynomial of a graph is considered. The calculation of some coefficients is shown to be # P-complete, whereas some other coefficients can be computed in polynomial time. However, even for a hard coefficient, it can be decided in polynomial time whether it is less than a fixed constant.