Evaluations of Topological Tutte Polynomials
نویسندگان
چکیده
We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, L(G), and of and Bollobás and Riordan, R(G), as well as for the classical Tutte polynomial T (G). For example, we express R(G) and T (G) as a sum of chromatic polynomials, show that R(G) counts non-crossing graph states and k-valuations, and reformulate the Four Colour Theorem in terms of R(G). Our main approach is to apply identities for the topological transition polynomial, one involving twisted duals, and one involving doubling the edges of a graph. These identities for the transition polynomial allow us to show that the Penrose polynomial P (G) can be recovered from R(G), a fact that we use to obtain identities and interpretations for R(G). We also consider enumeration of circuits in medial graphs and use this to relate R(G) and L(G) for graphs embedded in low genus surfaces.
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عنوان ژورنال:
- Combinatorics, Probability & Computing
دوره 24 شماره
صفحات -
تاریخ انتشار 2015