On the Matroid Intersection Conjecture

Ghaderi, Shadisadat

doi:10.33915/etd.5665

Cited by 2 publications

(4 citation statements)

References 26 publications

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“…By analysing their proof it is clear that the equivalence can be established if we restrict both conjectures to a class of matroids closed under certain operations. It allows us to prove Theorem 1.4 by showing the following instance of the Generalized Matroid Intersection Conjecture which itself is a common extension of the singular case by Ghaderi [16] and our previous work [21]: Theorem 1.4. If M and N are matroids in F ⊕ F * on the same countable edge set E, then they admit a common independent set I for which there is a partition…”

Section: Introductionmentioning

confidence: 88%

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On the Packing/Covering Conjecture of Infinite Matroids

Joó¹

2021

Preprint

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The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem by Edmonds and Fulkerson. A packing for a family (M i : i ∈ Θ) of matroids on the common edge set E is a system (S i : i ∈ Θ) of pairwise disjoint subsets of E where S i is panning in M i . Similarly, a covering is a system (I i : i ∈ Θ) with i∈Θ I i = E where I i is independent in M i . The conjecture states that for every matroid family on E there is a partition E = E p E c such that (M i E p : i ∈ Θ) admits a packing and (M i .E c : i ∈ Θ) admits a covering. We prove the special case where E is countable and each M i is either finitary or cofinitary. The connection between packing/covering and matroid intersection problems discovered by Bowler and Carmesin can be established for every well-behaved matroid class. This makes possible to approach the problem from the direction of matroid intersection. We show that the generalized version of Nash-Williams' Matroid Intersection Conjecture holds for countable matroids having only finitary and cofinitary components.

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Section: Introductionmentioning

confidence: 88%

“…Several partial results has been obtained for this generalization but only for well-behaved matroid classes. The positive answer is known for example if: M is finitary and N is cofinatory [6] or both matroids are singular 2 and countable [16] or M is arbitrary and N is the direct sum of finitely many uniform matroids [20].…”

Section: Introductionmentioning

confidence: 99%

On the Packing/Covering Conjecture of Infinite Matroids

Joó¹

2021

Preprint

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show abstract

“…From their proof it is clear that the equivalence can be established if we restrict both conjectures to a class of matroids closed under certain operations. It allows us to prove Theorem 1.2 by showing the following instance of the Generalised Matroid Intersection Conjecture which itself is a common extension of the singular case by Ghaderi [16] and our previous work [21]: Theorem 1.4: If M and N are matroids in F ⊕ F * on the same countable edge set E, then they admit a common independent set I for which there is a partition…”

Section: Introductionmentioning

confidence: 94%

“…Several partial results has been obtained for this generalisation but only for well-behaved matroid classes. The positive answer is known for example if: M is finitary and N is cofinitary [6] or both matroids are singular 2 and countable [16] or M is arbitrary and N is the direct sum of finitely many uniform matroids [20].…”

Section: Introductionmentioning

confidence: 99%

On the packing/covering conjecture of infinite matroids

Joó

2023

Isr. J. Math.

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The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family $$({M_i}:i \in \Theta)$$ ( M i : i ∈ Θ ) of matroids on the common edge set E is a system $$({S_i}:i \in \Theta)$$ ( S i : i ∈ Θ ) of pairwise disjoint subsets of E where Si is panning in Mi. Similarly, a covering is a system (Ii: i ∈ Θ) with $${\cup _{i \in \Theta}}{I_i} = E$$ ∪ i ∈ Θ I i = E where Ii is independent in Mi. The conjecture states that for every matroid family on E there is a partition $$E = {E_p} \sqcup {E_c}$$ E = E p ⊔ E c such that $$({M_i}\upharpoonright{E_p}:i \in \Theta)$$ ( M i ↾ E p : i ∈ Θ ) admits a packing and $$({M_i}.{E_c}:i \in \Theta)$$ ( M i . E c : i ∈ Θ ) admits a covering. We prove the case where E is countable and each Mi is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.

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On the Matroid Intersection Conjecture

Cited by 2 publications

References 26 publications

On the Packing/Covering Conjecture of Infinite Matroids

On the Packing/Covering Conjecture of Infinite Matroids

On the packing/covering conjecture of infinite matroids

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