The values of $m,n$ that makes $U_{m,n}$ graphic.

Question

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 $U_{m,n}.$

Answer 1

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.