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I am trying to find the values of $m,n$ that makes $U_{m,n}$ graphic. I am guessing that it is only graphic if $n = m + 1,$ I tried $U_{2,4},U_{2,5},U_{3,6}, U_{2,3},U_{4,5}$ and I think I am correct. Am I correct or I am missing something?

EDIT:

Here is the definition of $U_{m,n}:$

Let $m$ and $n$ be nonnegative integers with $m \leq n.$ Let $E$ be an $n$-element set and $\mathcal{B}$ be the collection of $m$ element subsets of $E.$ Then $\mathcal{B}$ is the set of bases of a matroid on $E$ and we denote this matroid by $U_{m,n}.$

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  • $\begingroup$ It would be helpful to say what $U_{m,n}$ denotes... $\endgroup$
    – Math1000
    Sep 24, 2023 at 21:27
  • $\begingroup$ @Math1000 sorry about that I edited my post $\endgroup$
    – Emptymind
    Sep 24, 2023 at 23:30
  • $\begingroup$ You gave the definition of a regular matroid, but to my knowedge $U_{m,n}$ generally denotes a uniform matroid. It is still confusing. $\endgroup$
    – Math1000
    Sep 25, 2023 at 5:38
  • $\begingroup$ @Math1000 you are correct I am so sorry I will edit my question $\endgroup$
    – Emptymind
    Sep 25, 2023 at 16:09

1 Answer 1

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Note that $U_{1,n}$ is the graphic matroid of an $n$-edge dipole graph, and the same is true for the corresponding duals - $U_{n-1,n}$ and the $n$-edge cycle graph.

The same is true for $U_{0,n}$, the graphic matroid of a graph with $n$ self-loops, and $U_{n,n}$, the graphic matroid of an $n$-edge forest.

Any other uniform matroid $U_{m,n}$ with $1<m<n-1$ contains $U_{2,4}$ as a minor and therefore is not graphic.

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  • $\begingroup$ Also, can you please recommend books for me (easy and clear ones) to study matroid theory. $\endgroup$
    – Emptymind
    Sep 25, 2023 at 20:28
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    $\begingroup$ I'm not particularly familiar with the topic but the texts by Oxley and Welsh seem decent enough. $\endgroup$
    – Math1000
    Sep 25, 2023 at 21:59

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