A direct and elementary proof of the well-definedness of the interior and exterior polynomials of hypergraphs

TL;DR

This paper offers a straightforward and elementary proof confirming that the interior and exterior polynomials of hypergraphs are well-defined, regardless of hyperedge ordering, extending Tutte's polynomial concepts.

Contribution

It provides a simple, direct proof of the well-definedness of hypergraph interior and exterior polynomials, avoiding complex polytope techniques.

Findings

Proof confirms independence from hyperedge ordering

Simplifies understanding of hypergraph polynomials

Extends Tutte polynomial concepts to hypergraphs

Abstract

T. K\'{a}lm\'{a}n (A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1/x,1)$ and $(1,1/y)$ to hypergraphs. The two polynomials are defined under a fixed ordering of hyperedges, and are proved to be independent of the ordering using techniques of polytopes. In this paper, similar to the Tutte's original proof we provide a direct and elementary proof for the well-definedness of the interior and exterior polynomials of hypergraphs.