What is a 4-connected matroid?
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912330020814848 |
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| author | Brettell, Nick Jowett, Susan Oxley, James Semple, Charles Whittle, Geoff |
| author_facet | Brettell, Nick Jowett, Susan Oxley, James Semple, Charles Whittle, Geoff |
| contents | The {\em breadth} of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. A matroid $M$ is {\em weakly $4$-connected} if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $λ(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is $4$-{\em connected} if $λ(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08832 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | What is a 4-connected matroid? Brettell, Nick Jowett, Susan Oxley, James Semple, Charles Whittle, Geoff Combinatorics 05B35 The {\em breadth} of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. A matroid $M$ is {\em weakly $4$-connected} if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $λ(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is $4$-{\em connected} if $λ(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set. |
| title | What is a 4-connected matroid? |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2310.08832 |