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Finite matchability under the matroidal Hall's condition

Attila Joó

Abstract

Aharoni and Ziv conjectured that if $ M $ and $ N $ are finitary matroids on $ E $, then a certain ``Hall-like'' condition is sufficient to guarantee the existence of an $ M $-independent spanning set of $ N $. We show that their condition ensures that every finite subset of $ E $ is $ N $-spanned by an $ M $-independent set.

Finite matchability under the matroidal Hall's condition

Abstract

Aharoni and Ziv conjectured that if and are finitary matroids on , then a certain ``Hall-like'' condition is sufficient to guarantee the existence of an -independent spanning set of . We show that their condition ensures that every finite subset of is -spanned by an -independent set.
Paper Structure (6 sections, 24 theorems, 15 equations, 2 figures)

This paper contains 6 sections, 24 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The main result
  4. Reductions via negligible and stable sets
  5. Equivalence classes of finite common independent sets
  6. A maximal directed set

Key Result

Theorem 1.1

A bipartite graph $G=(S,T,E)$ has a matching that covers $T$ iff there is no $X\subseteq T$ such that $N(X)$ can be matched to $X$ and no matching covers $X$.

Figures (2)

  • Figure 1: The negligibility of $G$.
  • Figure 2: The stability of $I$.

Theorems & Definitions (56)

  • Theorem 1.1: Infinite Hall's theorem by Aharoni, aharoni1991infinite
  • Theorem 1.2: Edmonds
  • Definition 1.3
  • Conjecture 1.4: aharoni1998intersection
  • Theorem 1.5
  • Lemma 2.3
  • Lemma 2.4: Edmonds, edmonds2003submodular
  • Lemma 2.5: joo2021MIC
  • Theorem 3.1
  • Definition 3.2
  • ...and 46 more