On Tutte polynomials and cycles of plane graphs.

*(English)*Zbl 0595.05033Let t(G,x,y) be the Tutte polynomial (or dichromate) of the connected plane graph G. It is known (Martin; Las Vergnas) 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; Rosenstiehl and Read).

Another result of Penrose 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)\(\neq 0\) or \(x=y=1\).

Another result of Penrose 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)\(\neq 0\) or \(x=y=1\).

##### Keywords:

Tutte polynomial; plane graph; number of crossing cycles; number of 3- edge colourings; cubic connected plane graph
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\textit{F. Jaeger}, J. Comb. Theory, Ser. B 44, No. 2, 127--146 (1988; Zbl 0595.05033)

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##### References:

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