On the evaluation at (3,3) of the Tutte polynomial of a graph (Q1121273)

The evaluation t(G;3,3) at (3,3) of the Tutte Polynomial (formerly called the dichromatic polynomial) of a planar graph G is given a combinatorial interpretation as a summation (over non-negative integers k) involving the numbers of Eulerian orientations of the medial graph H of G having exactly k saddle vertices. A divisibility property for t(G;3,3) is obtained as a corollary. A similar summation expresses the evaluation m(H;3) of the Martin Polynomial for a 4-regular graph H imbedded in an arbitrary surface; a similar corollary is obtained.