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Regular packing of rooted hyperforests with root constraints in hypergraphs

Pierre Hoppenot, Mathis Martin, Zoltán Szigeti

Abstract

The seminal papers of Edmonds \cite{Egy}, Nash-Williams \cite{NW} and Tutte \cite{Tu} have laid the foundations of the theories of packing arborescences and packing trees. The directed version has been extensively investigated, resulting in a great number of generalizations. In contrast, the undirected version has been marginally considered. The aim of this paper is to further develop the theories of packing trees and forests. Our main result on graphs characterizes the existence of a packing of $k$ forests, $F_1, \ldots, F_k$, in a graph $G$ such that each vertex of $G$ belongs to exactly $h$ of the forests, and in addition, each $F_i$ has between $\ell(i)$ and $\ell'(i)$ connected components and the total number of connected components in the packing is between $α$ and $β$. Finally, we extend this result to hypergraphs and dypergraphs, the latter giving a generalization of a theorem of Bérczi and Frank \cite{BF3}.

Regular packing of rooted hyperforests with root constraints in hypergraphs

Abstract

The seminal papers of Edmonds \cite{Egy}, Nash-Williams \cite{NW} and Tutte \cite{Tu} have laid the foundations of the theories of packing arborescences and packing trees. The directed version has been extensively investigated, resulting in a great number of generalizations. In contrast, the undirected version has been marginally considered. The aim of this paper is to further develop the theories of packing trees and forests. Our main result on graphs characterizes the existence of a packing of forests, , in a graph such that each vertex of belongs to exactly of the forests, and in addition, each has between and connected components and the total number of connected components in the packing is between and . Finally, we extend this result to hypergraphs and dypergraphs, the latter giving a generalization of a theorem of Bérczi and Frank \cite{BF3}.
Paper Structure (16 sections, 20 theorems, 51 equations)

This paper contains 16 sections, 20 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. General definitions
  4. Digraphs and dypergraphs
  5. Graphs and hypergraphs
  6. Packing
  7. Matroid theory
  8. Preliminary results
  9. Prior results
  10. New results
  11. Proofs
  12. Proof of Theorem \ref{['jjjnbbbnajkhypregdir']}
  13. Proof of Theorem \ref{['kjjhvhgu']}
  14. Proof of Theorem \ref{['kbjelciueundi']}
  15. Proof of Theorem \ref{['livzdljzvdmz']}
  16. ...and 1 more sections

Key Result

Theorem 1

The rank function $r$ of the sum matroid ${\sf M}$ of $k$ matroids ${\sf M}_i = (S, r_i)$ is given by the following formula

Theorems & Definitions (49)

  • Theorem 1: Edmonds, Fulkerson EF
  • Claim 1
  • proof
  • Lemma 1
  • proof
  • Claim 2
  • proof
  • Theorem 2: Kirkpatrick, Hell kirkpat
  • Theorem 3
  • proof
  • ...and 39 more