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Register a new userThe Las Vergnas Polynomial for embedded graphs
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The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas results on Eulerian circuits from graphs in surfaces of low genus to surfaces of arbitrary genus.
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We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem.
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