On the evaluation at (3, 3) of the Tutte polynomial of a graph

The dichromatic polynomial of a graph-now currently called the Tutte polynomial-was introduced by Tutte in 1954 as a generalization of chromatic polynomials studied by Birkhoff and Whitney. The generalization to matroids (combinatorial geometries) is due to Crapo in 1969. The Tutte polynomial of a matroid is relevant in a large number of problems involving numerical invariants of the matroid. We have introduced in [l, 31 a Tutte polynomial attached to the more general situation of a matroid perspective. Properties of this polynomial are particularly used in [Z] to study Eulerian cycles of 4-valent regular graphs imbedded in surfaces. In his thesis [S], Martin has introduced a polynomial in one variable related to the enumeration of Eulerian cycles for several classes of graphs (4-regular planar graphs, 4and 6-regular undirected graphs, 4-regular directed graphs). In each case this polynomial is defined by means of inductive relations obtained from reduction operations: it is shown that all sequences of reductions yield the same polynomial. We have generalized in [4] this polynomial-the Martin polynomial-to all graphs by using a different (algebraic) technique. The Tutte polynomial of a planar graph G is related to the Martin