A Whitney polynomial for hypermaps

Abstract

We introduce a Whitney polynomial for hypermaps. For maps, our definition depends only on the underlying graph and coincides with the usual definition, but for general hypermaps it depends on the topological structure. Our invariant satisfies a generalized deletion-contraction recurrence and it may be used to generalize the results of Arratia, Bollobás, Ellis-Monaghan, Martin and Sorkin connecting the circuit partition polynomial to the Martin polynomial of a graph. For hypermaps with hyperedges of length at most three our approach also allows generalizing most results connecting the chromatic polynomial and the flow polynomial with the Tutte polynomial of a graph.