Hypermaps with hyperedges of length at most $3$

TL;DR

This paper introduces methods for computing Whitney polynomials and enumerating spanning hypertrees in hypermaps with hyperedges of length up to 3, extending classical invariants to this hypergraph class.

Contribution

It develops deletion-contraction formulas and generalizes the reliability polynomial and random cluster model for hypermaps, providing explicit counts of spanning hypertrees.

Findings

Derived deletion-contraction formulas involving six types of loops and bridges.

Generalized the reliability polynomial and random cluster model to hypermaps.

Explicitly counted spanning hypertrees in reciprocals of plane graphs with degree at most 3.

Abstract

We study the computation of our recently introduced Whitney polynomial and the enumeration of the spanning hypertrees for hypermaps whose hyperedges have length at most $3$. This is a class of hypermaps where the computation of the above invariants depends only on the underlying (multi)hypergraph structure. We develop deletion-contraction formulas involving six types of generalized loops and bridges, and we prove results on special substitutions into our Whitney polynomial. We generalize the reliability polynomial and the random cluster model to hypermaps in general in such a way that they can be computed using our Whitney polynomial. Finally we explicitly count the spanning hypertrees in reciprocals of plane graphs in which every vertex has degree at most $3$.