A version of Tutte's polynomial for hypergraphs

TL;DR

This paper generalizes Tutte's polynomial to hypergraphs by defining hypertree activities, exploring their properties within polymatroids, and extending classical theorems to this broader context.

Contribution

It introduces a hypergraph version of Tutte's polynomial, linking hypertrees to polymatroid bases and establishing duality and planar representations.

Findings

Hypertrees form a lattice polytope as bases of a polymatroid.

The invariants extend to integer polymatroids.

A duality property generalizes Tutte's Tree Trinity Theorem.

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. In the definition, we associate activities to hypertrees, which are generalizations of the indicator function of the edge set of a spanning tree. We prove that hypertrees form a lattice polytope which is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. 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.