On Tutte polynomials and cycles of plane graphs

Abstract

Let t( G, x, y) be the Tutte polynomial (or dichromate) of the connected plane graph G. It is known ( Martin, Thesis, Grenoble, 1977; Las Vergnas, “Proceedings of the Conference on Algebraic Methods in Graph Theory,” Szeged, 1978) that t( G, x, x) can be expressed in terms of the family of partitions of the edge-set of the medial graph M( G) of G into non-crossing cycles. Moreover t( G, −1, −1) can be expressed in terms of the number of crossing cycles of M( G) ( Martin, Thesis, Grenoble, 1977, J. Combin. Theory Ser. B 24 (1978) , 318–324; Rosenstiehl and Read, Ann. Discrete Math. 3, 195–226). Another result of Penrose (“Combinatorial Mathematics and Its Applications,” D. J. A. Welsh, Ed., pp. 221–244, Academic Press, London/Orlando, 1971) on the number of edge-3-colorings of a cubic connected plane graph G can be viewed as an evaluation of t( G, 0, −3) in terms of the family of partitions of the edge-set of M( G) into cycles avoiding certain transitions. We unify and generalize these results by giving expressions of t( G, x, y) in terms of cycle partitions of M( G) for all x, y such that ( x − 1) ( y − 1) ≠ 0 or x = y = 1.