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A Cantor-Bernstein-type theorem for spanning trees in infinite graphs

Joshua Erde, Pascal Gollin, Atilla Joó, Paul Knappe, Max Pitz

Abstract

We show that if a graph admits a packing and a covering both consisting of $λ$ many spanning trees, where $λ$ is some infinite cardinal, then the graph also admits a decomposition into $λ$ many spanning trees. For finite $λ$ the analogous question remains open, however, a slightly weaker statement is proved.

A Cantor-Bernstein-type theorem for spanning trees in infinite graphs

Abstract

We show that if a graph admits a packing and a covering both consisting of many spanning trees, where is some infinite cardinal, then the graph also admits a decomposition into many spanning trees. For finite the analogous question remains open, however, a slightly weaker statement is proved.

Paper Structure

This paper contains 4 sections, 9 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Elementary proofs of Laviolette and Erdős-Hajnal
  3. A Cantor-Bernstein theorem for spanning trees in infinite graphs
  4. An open problem

Key Result

Theorem 1.1

Let $\lambda$ be an infinite cardinal. Then a graph admits a $\lambda$-decomposition if and only if it admits both a $\lambda$-packing and a $\lambda$-covering.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2: Laviolette, laviolette2005decompositions*Corollary 14
  • Theorem 1.3: Erdős and Hajnal, erdHos1967decomposition*Theorem 9
  • proof : Proof of Theorem \ref{['thm:laviolette']}
  • proof : Proof of Theorem \ref{['t:EHcover']}
  • Corollary 2.1
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof : Construction
  • ...and 3 more