13 problems

Let $\mathrm{M}$ be a matroid, and let $\{B_1,B_2\}$ and $\{B_1',B_2'\}$ be unordered pairs of bases such that the multiset union of $B_1$ and $B_2$ is equal to the multiset union…

Let $A$ and $B$ be bases of a matroid $\mathbf{M}$ with rank function $r$. For any $X\subseteq A\setminus B$ and $Y\subseteq B\setminus A$ and integer $p$ with $|Y| \le p \le |X|$,…

Let $M$ be a matroid of rank $r$, and let $(A,B)$ and $(A',B')$ be compatible basis pairs, meaning that … A symmetric exchange is an exchange of elements between the two bases that…

Let $\mathcal{M}$ be a matroid and let … be two bases. Gabow's conjecture. There exist permutations $\sigma$ and $\sigma'$ such that in the sequence … every set of $r$ cyclically c…

A matroid is subsequence-interchangeably base orderable (SIBO) if every pair of bases admits the subsequence exchange orderings specified in the paper. Sparse-paving SIBO conjectur…

A rank-$r$ matroid is subsequence-interchangeably base orderable (SIBO) if every pair of bases $A$ and $B$ admits orderings $a_1,\ldots,a_r$ of $A$ and $b_1,\ldots,b_r$ of $B$ such…

Let $M$ be a matroid of rank $r$, and let $X_1$ and $X_2$ be disjoint bases. The exchange distance of two basis sequences is the minimum number of symmetric exchanges needed to tra…

Weighted exchange-distance conjecture. The weighted exchange distance of $\mathbf{P}_1$ and $\mathbf{P}_2$ is at most

White's conjecture. The sequences $\mathcal{X}$ and $\mathcal{Y}$ are equivalent if and only if they are compatible. Compatibility is evidently necessary, and the conjecture is con…

Gabow's conjecture. There are orderings $A=(a_1,\dots,a_r)$ and $B=(b_1,\dots,b_r)$ such that

Common-basis conjecture. There exist partitions of $E\setminus\{x\}$ and $E\setminus\{y\}$ into $k$ pairwise disjoint bases that share a common basis.

Cyclic ordering conjecture. It is possible to place the elements of the ground set of $M$ on a circle so that every $r(M)$ consecutive elements form a basis of $M$.

Farber's conjecture. The basis pair graph of any matroid is connected.