Matroidal Cycles and Hypergraph Families

TL;DR

This paper introduces a new framework for hypergraphical matroids, generalizing previous concepts, defining hypergraph cycles and hypertrees, and characterizing hypergraphs related to matroid circuits.

Contribution

It proposes a novel definition of hypergraphical matroids applicable to all hypergraphs, unifying and extending prior models and introducing new notions of cycles and hypertrees.

Findings

Defined hypergraphical matroids for arbitrary hypergraphs

Established equivalence relations based on matroidal closures

Characterized hypergraphs isomorphic to circuit hypergraphs

Abstract

We propose a novel definition of hypergraphical matroids, defined for arbitrary hypergraphs, simultaneously generalizing previous definitions for regular hypergraphs (Main, 1978), and for the hypergraphs of circuits of a matroid (Freij-Hollanti, Jurrius, Kuznetsova, 2023). As a consequence, we obtain a new notion of cycles in hypergraphs, and hypertrees. We give an equivalence relation on hypergraphs, according to when their so-called matroidal closures agree. Finally, we characterize hypergraphs that are isomorphic to the circuit hypergraphs of the associated matroids.