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Matroidal Cycles and Hypergraph Families

Ragnar Freij-Hollanti, Patricija Šapokaitė

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.

Matroidal Cycles and Hypergraph Families

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.

Paper Structure

This paper contains 7 sections, 19 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Matroidal cycles
  3. Closures
  4. Hypergraphical matroids
  5. Matroidal trees
  6. Closure families
  7. Iterated derived matroids

Key Result

Proposition 2.1

Let $S$ be a matroidal cycle in a hypergraph $H$. Then there is a Berge cycle in $H$, containing a subset of the edges in $S$.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • ...and 49 more