A version of Tutte’s polynomial for hypergraphs

Abstract

Tutte’s dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. Our generating functions are sums over hypertrees, i.e., instances of a certain generalization of the indicator function of the edge set of a spanning tree. We prove that hypertrees are the lattice points in a polytope which in turn is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. Several properties are established, including a generalization of the deletion–contraction formulas. We also examine hypergraphs that can be represented by planar bipartite graphs, write their hypertree polytopes in the form of a determinant, and prove a duality property that leads to an extension of Tutte’s Tree Trinity Theorem.